As discussed in this question, the policy gradient algorithms given in Reinforcement Learning: An Introduction use the gradient \begin{align*} \gamma^t \hat A_t \nabla_{\theta} \log \pi(a_t \, | \, s_t, \theta) \end{align*} where $\hat A_t$ is the advantage estimate for step $t$. For example, $\hat A_t = r_t + \gamma V(s_{t+1}) - V(s_t)$ in the one-step actor-critic algorithm given in section 13.5.
In the answers to the linked question, it is claimed that the extra discounting is "correct", which implies that it should be included.
If I look in the literature to a seminal paper such as Proximal Policy Optimization Algorithms by OpenAI, they do not include the extra discounting factor, i.e. they use a gradient defined as \begin{align*} \hat A_t \dfrac{\nabla_{\theta}\pi(a_t \, | \, s_t, \theta)}{\pi(a_t \, | \,s_t, \theta_{\rm old})} \end{align*} which does not include the discounting factor (of course, it's dealing with the off-policy case, but I don't see how that would make a difference in terms of the discounting). OpenAI's implementation of PPO also does not include the extra discounting factor.
So, how am I supposed to interpret this discrepancy? I agree that the extra discounting factor should be present, from a theoretical standpoint. Then, why is it not in the OpenAI code or paper?