Difference quotient characterization conjecture for the Radon–Nikodým property of Banach function spaces

Let $X( Omega)$ be a Banach function space, and let the difference quotient criterion be the condition in which the estimate referenced as equation DQC1:1 characterizes membership in $W^1X( Omega)$. Difference quotient characterization conjecture. If the difference quotient criterion characterizes the space $W^1X( Omega)$, then $X( Omega)$ has the Radon–Nikod\fdm property. This proposes a converse to the known characterization of the Radon–Nikod\fdm property through difference quotients for vector-valued Sobolev spaces; whether the analogous implication holds for Banach function spaces is the question under consideration here.