Harbourne-Hirschowitz conjecture for plane linear systems

Let $m_1,\ldots,m_r$ be positive integers corresponding to general points $p_1,\ldots,p_r\in\mathbb{P}^2$. Let $\mathcal{L}=\mathcal{L}_d(m_1,\ldots,m_r)$ be the linear system of degree-$d$ plane curves with multiplicity $m_i$ at $p_i$, and let

$$e(\mathcal{L})=\max\left\{\binom{d+2}{2}-\sum_{i=1}^r\binom{m_i+1}{2}-1,-1\right\}.$$

A system is non-special when its actual dimension equals $e(\mathcal{L})$, and special otherwise. Let $\pi\colon V\to\mathbb{P}^2$ be the blow-up at the points $p_1,\ldots,p_r$. A curve $C\subseteq\mathbb{P}^2$ is a $(-1)$-curve if it is rational and its proper transform in $V$ has self-intersection $-1$. Harbourne-Hirschowitz conjecture. The system $\mathcal{L}$ is special if and only if its base locus contains a multiple $(-1)$-curve. This conjecture gives a geometric characterization of when general multiple points impose dependent linear conditions on plane curves; the source provides no evidence resolving it.