I'm not sure if Joshi's Corollary 3.12 is the same as Mochizuki's Corollary 3.12. He writes at the end of section 7.8 that "On the other hand, the above inequality suggests that the passage to the tensor product version $\hat{\hat{Θ}}_{\mathrm{Joshi}}^B$ should be expected to provide tighter upper bounds!" but he never proved the upper bounds, while Mochizuki's inequality specifically includes the upper bounds.