The SHGH vanishing conjecture for blow-ups of the projective plane

Let $X$ be the blow-up of $\mathbb P^2$ at $n$ generic points, and let $C$ be a curve on $X$. The SHGH vanishing conjecture asserts that

$$h^1(X,\mathcal O_X(C))=0$$

for every curve $C$ on $X$.

The source identifies this as an equivalent form of the SHGH conjecture in the stated setting when the constant in the preceding cohomology-bounding conjecture is $c_X=0$; its resolution status is not specified in the supplied text.