The F-conjecture for F-curves on moduli spaces of stable curves

Let $\overline{\rm{M}}_{g,n}$ be the moduli space of stable curves of genus $g$ with $n$ marked points. An F-curve is an irreducible component of the one-dimensional boundary stratum, and its cycle class is a curve class in $\overline{\text{NE}}_1(\overline{\rm{M}}_{g,n})$.

F-conjecture. The cone $\overline{\text{NE}}_1(\overline{\rm{M}}_{g,n})$ is generated by the cycle classes of F-curves. Equivalently, a line bundle on $\overline{\rm{M}}_{g,n}$ is nef if and only if its intersection number with each F-curve is nonnegative.

This is the same F-conjecture stated in the paper's preliminary discussion, expressing the expected characterization of nef line bundles by their intersections with F-curves. The source gives no evidence of a resolution.