In [M24] it is asserted that "considering $abc$ triples of the form $(1,p^n,1+p^n)$ for a prime number $p$ and an arbitrarily large positive integer $n$, one can verify that ABC/Szpiro inequalities can never be obtained as a result of summing up local inequalities at each prime of a number field."

Is this true, and what are references for this, or could you provide a sketch of the proof?

(Please note that I am disregarding all the controversial context and content of [M24], and focusing only on this mathematical statement).

[M24] S.Mochizuki, Report on the recent series of preprints by K.Joshi, march 24, 2024 https://www.kurims.kyoto-u.ac.jp/~motizuki/Report%20on%20a%20certain%20series%20of%20preprints%20(2024-03).pdf