Fulton's conjecture on the cone of curves of the moduli space of stable pointed curves

Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let $\overline{NE}_1$ denote the closure of the real cone of effective curves on $\overline{\mathbf{M}}_{0,n}$ modulo numerical equivalence. A vital curve is an irreducible component of the closed locus corresponding to a pointed genus-zero stable curve with at least $n-4$ nodes. Fulton's conjecture. The cone $\overline{NE}_1$ is generated by vital curves. Equivalently, a divisor is nef (respectively, ample) if and only if it intersects every vital curve nonnegatively (respectively, positively). This conjecture is the cone-theoretic form of the F-nef criterion and concerns the relationship between the Mori cone and the combinatorially defined vital curves; its resolution is not indicated in the supplied text.