SHGH conjecture for effective line bundles on blowups of the plane

Let $X_m$ be the blowup of $\operatorname{\mathbb{P}}^2$ at $m$ general points. A line bundle $\operatorname{\mathcal{O}}(D)$ is $(-1)$-special if $D$ is effective and $D.C \leq -2$ for some $(-1)$-curve $C$. SHGH conjecture. If $D$ is effective, then

$$\chi(\operatorname{\mathcal{O}}(D))=\dim\bigl(H^0(X_m,\operatorname{\mathcal{O}}(D))\bigr)$$

if and only if $\operatorname{\mathcal{O}}(D)$ is not $(-1)$-special. This is the cohomological prediction for effective linear series on blowups of the plane: speciality should arise exactly from $(-1)$-curves. The supplied status evidence says that the full conjecture is known for del Pezzo surfaces and for the blowup at nine general points, while the formulation is not resolved in general.