Harbourne-Gimigliano-Hirschowitz conjecture for plane linear systems

Let $p_1,\ldots,p_n$ be $n$ points on $\mathbb{CP}^2$ in general position, let $\pi:X\to\mathbb{CP}^2$ be their blowup, let $H$ be the divisor class of a line, and let $E_i$ be the exceptional divisor over $p_i$. For degree $d$ and multiplicities $m_1,\ldots,m_n$, set

$$D=d\pi^*H-\sum_{i=1}^nm_iE_i.$$

Harbourne-Gimigliano-Hirschowitz conjecture. The linear system $|D|$ has expected dimension

$$\max\left(-1,\frac{d(d+3)}2-\sum_{i=1}^n\frac{m_i(m_i+1)}2\right)$$

if and only if there is no $(-1)$-curve $C$ on $X$ such that

$$C\mathbin{.}D\leq -2.$$

This conjecture predicts that speciality of plane linear systems with general assigned points is explained precisely by $(-1)$-curves. It is closely related to Nagata's conjecture and remains open in the stated generality.