Fulton's weak nefness conjecture for the moduli space of stable pointed curves

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves. A divisor $E$ on $\overline{\mathcal{M}}_{0,n}$ is $F$-nef if $C(a,b,c,d).E\geq 0$ for every vital curve corresponding to a partition with part sizes $a,b,c,d$. Fulton's conjecture. The $F$-nef divisors are nef: if $C(a,b,c,d).E\geq 0$ for all $(a,b,c,d)$, then $C.E\geq 0$ for every irreducible curve $C$. This weak form is proved for $n\leq 7$; for $n\geq 8$, no counterexample is known, but no proof is known either.