Reduced systems conjecture for blow-ups of Hirzebruch surfaces

Let $\mathbb{F}_{e,r}$ be the blow-up of a Hirzebruch surface $\mathbb{F}_e$ at very general points, and let $\widetilde{C}_e$ be the strict transform of its negative section. A non-empty linear system $\mathcal{L}$ is reduced when its general curve is reduced, and $\widetilde{C}_e$ is not a fixed component when every member does not contain it as a fixed component. Reduced systems conjecture. If the general curve of a non-empty linear system $\mathcal{L}$ on $\mathbb{F}_{e,r}$ is reduced and $\widetilde{C}_e$ is not a fixed component of $\mathcal{L}$, then $\mathcal{L}$ is non-special. The paper states that Laface's conjecture implies this conjecture, but the supplied text gives no resolution of either claim.