Segre's conjecture on special plane linear systems

Let ${\mathcal{L}}_{2,d}(-\sum_{i=1}^hm_iP_i)$ be a linear system of plane curves with general multiple base points, where $P_1,\dots,P_h$ are general points and $m_1,\dots,m_h$ are positive integers. The system is special when its dimension is strictly greater than its expected dimension.

Segre's conjecture. If ${\mathcal{L}}_{2,d}(-\sum_{i=1}^hm_iP_i)$ is special, then its general member is non-reduced; equivalently, by Bertini's theorem, the system has a multiple fixed component.

This conjecture proposes that speciality of plane systems with general multiple points is always explained by non-reduced fixed components. Its status is not specified in the supplied text.