And Second, there is Mochizuki's Section 2. This reduces to the previous case with the help of the paper [GenEll], where in particular a valid choice of $\ell$ from Section 1 is ensured (among other reductions to "generic arithmetic elliptic curves"). There the inequality that figures is, literally (?!), but it is asserted up to finitely many exceptions covered by [GenEll] (in principle, their height is bounded above). So, by itself, this is no contradiction.

Postscriptum: I commented upon two distinct parts of Mochizuki's IUTT-IV - my apology for the ambiguity. First of all I have in mind his section 1: this is completely independent of the paper [GenEll], and contains an unconditional inequality - in Thm, 1.10 - in which figures the auxiliary prime $\ell$ (then take $\varepsilon \sim 1/\ell$ and you get (?!)), but valid only under the assumption that $\ell$ is generic for $E$, in the sense mentioned above. It is here that I felt there might be a contradiction. [cont]

(cont) ... such that there is some prime in $(1/\varepsilon, 2/\varepsilon)$ not dividing neither $abc$ nor any of its prime exponents, then (!?) holds. (With absolute constants $A,B < \infty$). This would be extremely disturbing.

Here is a final post. Working backwards (taking big enough $\ell$ so that the conjectured "uniform open image theorem" holds for $\ell$, and considering only $abc$-triples which, together with all of their prime exponents, are not divisible by $\ell$), we can see that: Assuming the uniform open image conjecture, Section 1 of IUTT-IV (see Theorem 1.10 and the backward-references there) implies, as written that there is an absolute $\epsilon_0 > 0$ such that, for any $0 < \varepsilon < \epsilon_0$, and any $abc$-triple such that (cont)

Indeed, the $K_{\varepsilon}$-constant is insignificant as far as the qualitative statement of ABC goes: one can just say that $c < \mathrm{rad}(abc)^{1+\varepsilon}$ has finitely many exceptions. But there is also interest in the form of $K_{\varepsilon}$: for example, there is A. Baker's conjecture: $c \leq A \cdot \prod_{p \mid abc} (p/\varepsilon)^{1+\varepsilon}$ for all $0 < \varepsilon leq 1$, with an absolute constant $A$. In any case, Mochizuki has an extremely sharp version of the ABC-conjecture. It is claimed to be effective, too.

... which certainly seems disturbing: assuming a uniform Serre's "open image theorem," it is then enough to take any big enough prime $\ell$ not dividing $abc$ as well as the prime exponents of $abc$. I stop here: I just wished to voice this as potential counterexample to Mochizuki's final estimate in section 1 of his fourth paper. My apology if I have got it wrong.

Here is what, to my understanding, Mochizuki claims in his Theorem 1.10 on pp. 22-23 (see the final display before the proof on p. 23, and also see IUTT-I, p. 50 for the assumptions on the "initial $\Theta$-data" $(\mathbb{Q},E,\ell)$, where $E$ is the associated elliptic curve $y^2 = x(x-a)(x+b)$ needed to make the translation to (?!)): For an arbitrary triple $(a,b,c)$, consider a prime $\ell$ which is generic for the associated elliptic curve $E$, in the sense that the assumptions of the third paragraph above are fulfilled. Then (?!) holds with $\varepsilon := 29/\ell$.

Certainly Mochizuki does not claim that (!?) holds for all $(a,b,c)$; he claims instead that, for a given $\varepsilon$, he can effectively determine all counterexamples to (!?) using his (entirely elementary!) paper [GenEll]. This by itself is no contradiction; but it is very striking claim. See also his Remark 2.3.2 in IUTT-IV.