Relation between flatness of Frobenius pushforward and Kähler differentials in characteristic p (outside the F-finite case)

Determine the relationship between the flatness of the Frobenius pushforward F_*A and the flatness of the module of Kähler differentials Omega_{A/F_p} as A-modules for Noetherian reduced F_p-algebras that are not F-finite; in particular, ascertain whether the flatness of F_*A is equivalent to the flatness of Omega_{A/F_p} in this generality.

Background

The paper highlights a question posed by Javier Carvajal-Rojas concerning the interplay between flatness of the Frobenius pushforward F_*A and flatness of the module of Kähler differentials Omega_{A/F_p} for F_p-algebras. In the Noetherian, F-finite, reduced setting, both modules are finite over A and their flatness is known to be equivalent; in contrast, for non-reduced rings, there are simple counterexamples (e.g., Omega_{A/F_p} can be free while F_*A is not flat).

The author explicitly notes that outside the F-finite hypothesis, even for Noetherian reduced rings, an answer is not known. The paper develops results in a different direction (valuation rings and related approximations), but this specific question remains unresolved in the classical Noetherian reduced context beyond F-finite algebras.