I apologize for the premature comment (I should have worked out many more details before posting), 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. Those are 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 - equiivalently, as it were - elliptic curves] come from an explicit bound on the [Faltings] height.

In fact, if you look at the key Theorem 1.10 (which is independent of the theory of [GenEll]), you will see an explicit version of Szpiro's inequality [for the associated elliptic curve $E: \, y^2 = x(x-a)(x+b)$] equivalent to (!?), asserted under the sole assumption that there is a not too large prime $\ell$ [say, a choice on the order of $\ell \sim 1/\epsilon$ would yield precisely (!?)!] 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 (!?) 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 (!?) may not hold without the - seemingly mild - assumptions of the previous paragraph).

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.

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 (!?)!] 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 (!?) 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 (!?) may not hold without the - seemingly mild - assumptions of the previous paragraph).