Segre's conjecture on reduced linear systems

Let $S$ be a smooth projective surface, and let $\mathcal L$ be a linear system on $S$. The system is reduced if it has no multiple fixed components, and it is non-special when its dimension equals its expected dimension.

Segre's conjecture. If $\mathcal L$ is non-empty and reduced, then it is non-special.

By Bertini's second theorem, this would imply that every special system has an irreducible curve $C$ whose doubled divisor is contained in the base locus. Thus the conjecture would reduce the classification of special systems to the classification of such fixed curves; it is stated here as an extension of Segre's conjecture from plane linear systems to arbitrary surfaces.