The SHGH conjecture for negative curves on blowups of the plane

Let $X$ be a surface obtained by blowing up $n$ general points of $\mathbf{P}^2$. A curve $C\subset X$ is exceptional in the sense used in the paper when it is one of the distinguished negative curves arising from the blow-up, and a divisor $F$ is nef when $F\cdot C\ge0$ for every curve $C$.

SHGH negative-curve formulation. Every reduced irreducible curve $C\subset X$ with $C^2<0$ is an exceptional curve, and for every nef divisor $F$ either

$$h^0(X,F)=0$$

or

$$h^1(X,F)=0.$$

This is the appendix formulation of the SHGH conjecture and is intended to be equivalent to the Hilbert-function formulation. It remains open for blow-ups at sufficiently many general points.