Segre–Harbourne–Gimigliano–Hirschowitz conjecture for plane blow-ups

Let $X$ be the blow-up of $\mathbb P^2_\mathbb C$ in $n$ points in general position, and let $H$ be the pullback of a hyperplane while $E_i$ are the exceptional divisors. For integers $d>0$ and $m_i\geq 0$, set

$$D=dH-\sum_{i=1}^n m_iE_i.$$

Segre–Harbourne–Gimigliano–Hirschowitz conjecture. Either

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

or there exists a $(-1)$-curve $C\subseteq X$ such that $C\cdot D\leq -2$. This predicts the dimension of the linear system of plane curves with prescribed multiplicities at general points; its general validity remains open.