Category analogue of sup-measurability problem

A function F: R2→ R is called sup-measurable if Ff : R→ R given by Ff(x) = F(x,f(x)), x ∈R, is measurable for each measurable function f: R→ R. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analogues. In this paper we will show that the existence of the category analogues of sup-measurable non-measurable functions is independent of ZFC. A similar result for the original measurable case is the subject of a work in prepartion by Rosłanowski and Shelah.