The $-1$-curves conjecture for blowups of the projective plane

Let $S=S_n$ be the blowup of the projective plane at $n$ very general points. Let $N^1_{\mathbb R}(S)$ be the real Néron–Severi space, let $\overline{\operatorname{NE}}(S)$ be the Mori cone, and let $\mathscr L$ denote the light cone. A $-1$-curve is a curve $c\subset S$ isomorphic to $\mathbb P^1$ with $c^2=-1$.

$-1$-curves conjecture. The Mori cone of $S$ is generated by the light cone and the $-1$-curves. Dually,

$$\operatorname{Nef}(S)=\{v\in\mathscr L\mid v\cdot c\ge 0\text{ for every }-1\text{-curve }c\subset S\}.$$

This conjecture concerns the geometry of the Mori and nef cones for blowups at very general points, particularly when $n>9$, where these cones are not completely understood. The source attributes it to Tommaso de Fernex and cites an erratum, but gives no resolution status.