Hirschowitz's conjecture on special plane linear systems

Let $\mathcal{L}=\mathcal{L}(d;m_0,m^n)$ be a linear system of plane curves with one base point of multiplicity $m_0$ and $n$ base points of equal multiplicity $m$. A system is $(-1)$-special if there are $(-1)$-curves $A_1,\ldots,A_r$ such that $\mathcal{L}\cdot A_j=-N_j$ with $N_j\geq 1$ for every $j$ and $N_j\geq 2$ for some $j$, while the residual system $\mathcal{M}=\mathcal{L}-\sum_jN_jA_j$ has non-negative virtual dimension and non-negative intersection with every $(-1)$-curve. Hirschowitz's conjecture. Every special system is $(-1)$-special. This is the main conjecture concerning homogeneous and quasi-homogeneous plane linear systems with assigned base-point multiplicities. The source states it as a restatement of a conjecture of Hirschowitz; no resolution is indicated here.