The flat-act cover conjecture for right-reversible monoids

Let $S$ be a monoid, and write $\operatorname{Act}$-$S$ for the category of right $S$-acts. The monoid $S$ is right-reversible when any two principal right ideals intersect. Let $\mathcal{F}_S$ denote the class of flat right $S$-acts, and let FCC denote the assertion that every right $S$-act has a flat cover. Flat-act cover conjecture. FCC holds for $\operatorname{Act}$-$S$ whenever $S$ is right-reversible. The paper proves FCC under the additional assumption that flat $S$-acts are closed under stable Rees extensions; the conjecture asks whether that assumption is superfluous.