https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=1

Mochizuki considers infinitely many distinct isomorphic copies of $\pi_1(X)$’s, but could not tell us what goes wrong if we simply identify all of them with one another, and with $\pi_1(X_0)$ for some fixed $X_0$ — there is no diagram that commutes in his situation but does not commute under this further identification. (In my manuscript with Stix, we simply went through Mochizuki’s argument with this further identification, pinpointing what goes wrong. If this further identification causes problems, just tell us which diagram it is whose commutativity is rescued by not explicitly identifying $\pi_1(X)$’s.)