If the computational components of the forward feed through the network have no curvature, which is normally the case in a sum of products, then it can be proven that any constant pixel value is equivalent in terms of effect on convergence results. We wouldn't expect a proof for that, since it would be too trivial to spend time writing up for publication. In general, functioning vision systems have feed forward computational components with curvature, so the padding is likely significant.
Even the convolutional layers may have activation functions or something even more complex going forward, as noted in Gauge Equivariant Convolutional Networks and the Icosahedral CNN (Taco S. Cohen, Maurice Weiler, Berkay Kicanaoglu, Max Welling, 2019).
If purely stochastic values with value distributions like that of the un-padded coordinates are used, it may be possible to prove that some gain is made, but none appeared in a few academic article searches just made. Not surprisingly, there are many proofs regarding the properties of various message padding strategies for cryptography.
Short of the inclusion of thermal or quantum noise acquisition devices in VLSI circuitry and exposure of those devices in software, purely stochastic values cannot be generated. This leaves the risk of a learning approach expected to extract features from frames learning features of the pseudo-random noise generator used to pad.
The answer is that none are universally correct and there appears to be much work to do in proving advantages between different techniques in as many cases as such advantages can be proven.