For concreteness, consider the $abc$-inequality with $\epsilon=1$, i.e. that there is some integer $K$ such that for all coprime integers $a,b,c$ with $a+b=c$, one has

$$abc\leq K \prod_{p|abc} p^6.$$

Mochizuki makes the true and very simple observation that this can never be proved by proving it "one prime at a time", i.e. proving that for every prime $p$, it divides the left-hand side at most as often as the right-hand side, up to some error term $K_p$ depending only on $p$ (and not on $a,b,c$). Indeed, the right-hand side is only divisible $6$ times by any large enough $p$, while the left-hand side can become arbitrarily divisible, as the example $(1,p^n,1+p^n)$ pointed out by Mochizuki shows.

This observation is relevant to the discussion of Joshi's manuscript. Indeed, the key result is Theorem 6.10.1 in Joshi's paper, giving upper and lower bounds on a certain volume. Both inequalities are proved by summing up local inequalities, i.e. inequalities at each prime $p$. For the first one, see the proof of Theorem 9.11.1 of Joshi's previous paper. For the second one, see the proof of Theorem 6.10.1. I believe the actual mistake is Proposition 6.10.7, whose proof simply refers to Mochizuki's work. However, if one notes that the proof of Theorem 9.11.1 also proves a corresponding lower bound in the context of Proposition 6.10.7, one sees that these two inequalities are contradictory, using the template of the simple argument above.

Let me stress that this argument is extremely robust: No matter the details of the inequalities around (and possibly other auxiliary hypotheses in place), the $abc$ inequality can not be obtained in this way. But both the proofs of Theorem 9.11.1 and Theorem 6.10.1 clearly indicate that they are obtained in this way. For all I can see, this means that Joshi's proof strategy is irreparably flawed.

I sent an e-mail to Joshi, asking for clarification, on the day he posted his manuscript to the arXiv. While he did kindly answer my e-mail, I have not yet received clarification on this point.

Let me end by saying that I wished that Mochizuki had focused his manuscript only on this mathematical point, and that I feel deeply sorry for Joshi for the rest of the manuscript (and more generally, I feel deep embarassment as a member of the mathematical community).