A proof of the plausibility of AGI is going to require a more formal definition of AGI than the one proposed in The AI Revolution: The Road to Superintelligence by blogger Tim Urban (2/10/2015 updated 4/12/2015), which was neither peer reviewed nor supported by research or statistical validation of any kind. 
A Proposed Definition of General Intelligence
The idea of computer intelligence has broadened since the period when IQ testing was popular and the General Problem Solver computer program was created (1959 by Herbert A. Simon, J. C. Shaw, and Allen Newel). John Bradshaw published Reclaiming Virtue, a foray into the idea of moral intelligence. Emotional intelligence has been discussed in numerous books and articles. Neural nets, Bayesian approaches, and fuzzy logic have emerged and been used successfully in decision making systems.
Alternatives have been offered to the Turing Test, pointing out that Alan Turing's Imitation Game (a human observer not being able to determine from a blindfold conversation that a computer conversant was not human) was not intended to be the sole metric in validating artificial intelligence. 
In light of advancements, a good definition of general intelligence must
- Match the intuitive understanding of general as an adjective;
- Match the intuitive understanding of intelligence as a noun;
- Distinguish general intelligence from systems that cannot learn new approaches (such as natural language query systems like Siri or optimizing decision making software like cargo routers);
- Be able to reliably and unambiguously attain an arbitrary goal provided the goal is well defined (arbitrary meaning that nothing about the goal can be known at the time of program development or its execution); and
- It must learn about the domains required to attain the goal after the program is running 3, 4.
The goal could be
- Get the phone number of a particular person at a bar through conversation only;
- Determine if a closed form can be found for a set of differential equations, one cannot be, or it cannot be determined if one can be found;
- Develop a new way to acquire electrical energy from sunlight; or
- Any other goal.
This is a proposed solution that encapsulates and summarizes all of these conditions.
Let General Intelligence be defined as, "The ability to adapt to the presence of arbitrary obstacles during the attainment of a specific goal and reach the goal nonetheless if it is at all possible to do so."
An immediate reaction of a thoughtful person might be, "That is more than general intelligence," but if you drop the word arbitrary, then one cannot claim that the intelligence is general, since it would clearly be limited by at least one case. If once case, it would be difficult to prove that an infinite number of classes of cases would also be opaque to the therefor limited intelligence.
Another critique might be that the, "At all possible to do so," would imply perfection or even deity. But if the goal is attainable, a person might tire, but why would a computer tire? Persistence is trivial for a computer.
The time to the solution might be infinite, but we are not defining Fast General Intelligence. We are defining General Intelligence, and once that is attained, speeding it up is a matter of optimization, process distribution, software scalability, and concurrence.
Another critique might be that the definition is absent of the word Artificial, but it is best without that word. If our intention is to automate things that humans do which computers cannot yet do, or exceed or extend human intelligence, then the definition's application should be independent of the tested system. The definition should not distinguish between what would essentially be prejudicial criteria with regard to the scientific treatment of intelligence and its general abilities.
- Whether it is programmed via compiled code, scripted code, some other form of programming, or via DNA
- Whether parents are involved in the development of domain knowledge
- Whether an education system is involved in the development of domain knowledge
- Whether the intelligence has access to muscles or some other environmental control
- Whether the intelligence has an eye or a camera
- Whether the intelligence is terrestrial
- Whether the intelligence is taught or self-taught
Existence Proof or Disproof
If this definition is acceptable, then clearly there is no proof that the human being exhibits General Intelligence. An intelligent system may not be able to either attainment an arbitrary goal or determine that the goal is not achievable for any of many reason including these.
- Attainment requires mental energy (patience and persistence) not possessed.
- Attainment requires cognitive skills outside the capabilities of the processing hardware or neural system.
- Attainment is blocked by closed mindedness in some topic area.
- Attainment is blocked by unwillingness to perform some class of operations.
- Attainment requires overcoming a DNA based susceptibility such as a mental disorder (anxiety disorder, addiction).
- Attainment is detoured by a cognition that leads thought or action down a blind alley and is without remedy.
- Attainment is blocked by the impossibility of the goal and pride blocks admission of failure, the admission of which would actually be within the possible valid results for General Intelligence.
- Attainment requires some non-deterministic yet non-random element used to generate intent or select from among choices (such as a soul or karma).
- Attainment requires more time than the life of the system attaining it.
- Attainment has another time limit that cannot possibly be met with the computational resources available.
There is no rigorous proof that General Intelligence as per the proposed definition is possible by all Turing Machines, but it is a worthy research problem, since all of the above listed failure causes except probably the requirement of some non-deterministic yet non-random element are likely to be overcome by computers.
The above list however, is a fairly rigorous proof that human beings do not exhibit reliable general intelligence. The human impulse and current popular belief, humanism, rebels against this idea, but it is nonetheless true.
The last question then reverts to, "How can we know that everything that can be programmed in a Turing Machine can be accomplished by a human brain?" The answer to that question is answered by this goal, inspired by the first and last attainment failure causes listed above: "You are stuck on an island with 1,000 pads of paper and 1,000,000 sharpened pencils with food and water and shelter but no computer or calculator, and the goal is to calculate pi to 1,000,000 decimal places."
Neurons can simulate neurons, but there is the question of accuracy in the simulation and the chaotic aspect of an astronomically complex nonlinear system. Various theories of existence or non-existence have been proposed for arbitrary goal attainment, but not all cases have been proved or many assumptions were required to gain the proof such that the goal statement is not truly arbitrary.
There is the possibility of some non-deterministic yet non-random element used to generate intent or select from among choices (such as a soul, autonomous or given purpose, meaning by fiat, or karma) which is neither proven nor dis-proven scientifically (although many will assert without actually providing a rigorous proof that it has been so). This is the key to answer the question originally posed.
In examination of the context of the question more closely, it is worthwhile to note that a Turing Machine is a machine able to execute an arbitrary set of arbitrarily linked deterministic operations. Such machines are not unlimited. They can create pseudo random numbers but non-deterministic phenomena cannot result from deterministic operations, thus a random number cannot be generated by a Turing Machine. Even a non-deterministic but meaningful choice cannot be made by a Turing Machine, yet a brain may be able to perform either.
Stochastic quantum mechanics is widely accepted among physicists and it is possible that some things are unknowable but measurable, some things may be knowable but immeasurable, and there may even be immeasurable and unknowable things. It is possible that something undiscovered or, for some reasons, immeasurable that human brains posses have that is beyond the Turing Machine.
This is also a possible area for further research, however it is rarely researched because the study of immeasurable phenomena, although such phenomena may exist, cannot be easily studied because they are immeasurable.
John von Neumann (perhaps brighter than Newton, Einstein, Planck, and Hawking) was correct in differentiating the fundamentals of a computer and a brain. Although there is surely some overlap in the demonstrated abilities of humans and computers, neither may ever be a subset of the other. Futurists may disagree, but it would be an opinion, not a proof.
Notes and References
 I don't see any evidence online that the author of The AI Revolution: The Road to Superintelligence, Tim Urban, is a computer scientist or holds a relevant degree. If you look at the article, you can see that the graphs given are invented trends, not driven by any real world data. It is essentially science fiction — popular and entertaining, but not rational conclusions drawn from either repeatable experiments or randomized studies.
 Testing if a computer has human-level intelligence: Alternative to 'Turing test' proposed Science News, 11/19/2014, Georgia Institute of Technology
 If the statement of the arbitrary goal precedes domain knowledge acquisition, some or all of the intelligence may reside outside the running program, contained in whatever mechanisms or people interact with the program or its data after the goal is introduced.
 This constraint does not exclude the possibility of asking other intelligent sources for information without delegating decisions about approaches to overcoming obstacles to these external helpers.