>>5 南出論文について ”In fact, the estimate in the first display of Corollary C may be strengthened roughly by a factor of 2 by applying the [slightly less elementary] results of [Ink1], [Ink2] [cf. Remarks 5.7.1, 5.8.2].”（下記） とあるので、「まだ改良の余地あり」と読みました
www.kurims.kyoto-u.ac.jp/~motizuki/Explicit%20estimates%20in%20IUTeich.pdf Explicit Estimates in Inter-universal Teichmuller Theory. PDF NEW!! (2020-11-30) いわゆる南出論文 P5 The proof of Corollary C is obtained by combining • the slightly modified version of [IUTchI-IV] developed in the present paper with • various estimates [cf. Lemmas 5.5, 5.6, 5.7] of an entirely elementary nature.
In fact, the estimate in the first display of Corollary C may be strengthened roughly by a factor of 2 by applying the [slightly less elementary] results of [Ink1], [Ink2] [cf. Remarks 5.7.1, 5.8.2]. [The authors have received informal reports to the effect that one mathematician has obtained some sort of numerical estimate that is formally similar to Corollary C, but with a substantially weaker [by many orders of magnitude!] lower bound for p, by combining the techniques of [IUTchIV], §1, §2, with effective computations concerning Belyi maps. On the other hand, the authors have not been able to find any detailed written exposition of this informally advertized numerical estimate and are not in a position to comment on it.]
