Thanks for the reference - I hadn't seen it! I see now that I should frame the question, more generally, to address rational self-maps $\varphi : X \dashrightarrow X$ such that the orbit of $x_0$ lies in the domain of $\varphi$.

@Will Jagy: Thanks for the suggestion, as well as for the links! I just made the latest - and probably last - revision to write down the estimate that Mochizuki expects (which is corrected and confirmed by him), and to say that it can definitely not be disproved in a similar way as the current Thm 1.10 was (thus, it does now seems reasonable). I don't think I will make any more revisions to this post at least until Mochizuki uploads his corrections to IUTT-III and IUTT-IV. (Unless one could solve the open question that I outlined to probe the corrected inequality; but this seems unlikely).