Cellular generation conjecture for flat-pure monomorphisms

Let $S$ be a monoid, and let $\mathcal{F}$-PureMono be the class of pure monomorphisms of right $S$-acts whose Rees quotient is flat. A class of monomorphisms is cellularly generated if it is generated by a set of morphisms under the relevant cellular constructions. Cellular-generation conjecture. If $S$ is right-reversible, then $\mathcal{F}$-PureMono is cellularly generated. This is presented as a stronger conjecture than the assertion that every right $S$-act has a flat cover for right-reversible $S$; its resolution is not given.