I apologize for the premature comment (I should have worked out more details before posting), and also if this answer is inappropriate here, but it almost seems as if Mochizuki's claimed inequality (see the display before the proof on p. 23; and then that on p. 36) - which is explicit, and purely effective - might be too strong to be true. In remark 2.3.2 on page 40 of IUTT-IV, Mochizuki himself almost acknowledges the possibility of finding a counterexample (Cf. the theorem of Stewart-Tijdeman). This tension, it seems, is yet to be cleared up.
Perhaps my intuition is completely failing me: it would be most remarkable to have such a strengthening of the conventional $ABC$-conjecture. Namely: the mentioned bounds, translated to the most common enunciation of $ABC$ (with rational integers $a+b = c$, $(a,b) = 1$), look like $$ (!?) \hspace{3cm} c < A \varepsilon^{-B} \cdot \mathrm{rad}(abc)^{1+\varepsilon}, $$ with $A,B < \infty$ numerical constants. This is supposed to hold, for each $\varepsilon \in (0,\epsilon_0)$, outside of an "exceptional set" that is effectively controlled entirely, it seems, by the elementary theory of Mochizuki's paper [GenEll] ("Arithmetic elliptic curves in general position") - in which all exceptional sets [of - equivalently, as it were - elliptic curves] come from an explicit bound on the [Faltings] height.
[EDIT: On second thought, I overlooked a subtle point: the Szpiro inequality $\log|\Delta_{\mathrm{min}}(E)| \leq (6+\varepsilon) \log{N_E} + O_{\varepsilon}(1)$, with which Mochizuki works initially, only implies the weaker, "averaged" $ABC$-conjecture $abc < K_{\varepsilon} \cdot \mathrm{rad}(abc)^{3+\varepsilon}$; thus, in discussing the implications of Mochizuki's Thm. 1.10, I should replace (!?) by $$ (\textrm{Weaker}!) \hspace{3cm} abc < A \varepsilon^{-B} \cdot \mathrm{rad}(abc)^{3+\varepsilon}. $$ ]
In fact, if you look at the key Theorem 1.10 (which is independent of the theory of [GenEll]), you will see asserted an explicit version of Szpiro's inequality [for the associated elliptic curve $E: \, y^2 = x(x-a)(x+b)$], under the sole assumption that there is a not too large prime $\ell$ [say, a choice on the order of $\ell \sim 1/\varepsilon$ would yield precisely (!Weaker)!] such that the triple $({ \mathbb{Q}},E,\ell)$ satisfies the generic "initial $\Theta$-data" assumptions of [IUTT-I, Section 3, p. 50] needed for the general theory: essentially, that the representation $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\mathbb{Z}/ \ell)$ on $E[\ell]$ is a surjection, and $\ell \nmid abc$ [i.e., $\ell$ does not divide the degenerate places of $E$], and $\ell$ does not divide any of the exponents in the prime factorization of $abc$ [i.e., $\ell$ is prime to the orders of the $q$-parameters at the $\mathbb{G}_m$-reductions of $E$].
All by itself, the dependence on $\varepsilon$ of the constant on the right-hand side of (!?), or (!Weaker), is very far from a plausible statement of the $ABC$ conjecture. Instead, A. Baker has conjectured that $B$ should be taken on the order of the number of prime factors in $abc$ (while $A$ could be kept an absolute constant). The result of Stewart-Tijdeman [cf. Bombieri-Gubler, Heights in Diophantine Geometry p. 419] implies that Baker's conjecture is close to optimal. (In particular, that (?!), (!Weaker) may not hold without the - seemingly mild - assumptions of the previous paragraph).
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" $(\mathbb{Q},E,ℓ)$, where $E$ is the associated elliptic curve $y^2=x(x−a)(x+b)$ needed to make the translation to (?!)): For an arbitrary $abc$-triple, 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 (!Weaker) holds with $\varepsilon:= 1/\ell$. This certainly seems disturbing: assuming a uniform Serre's "open image theorem," it is enough to take any big enough prime $\ell$ not dividing $abc$ as well as the prime exponents of $abc$. Working backwards, we 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 there is a prime in $\big( 1/\varepsilon, 2/\varepsilon \big)$ not dividing neither $abc$ nor any of the prime exponents of $abc$, then (?Weaker) holds for $(a,b,c)$ (with absolute constants $A,B < \infty$).
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 feel there might be a contradiction.
And second, there is Mochizuki's Section 2. This reduces the general case of the $ABC$ conjecture to the restricted version covered in Section 1; the reduction is entirely taken care of by the elementary 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, as noted in the comments.