Laface's special-divisor conjecture for blow-ups of Hirzebruch surfaces

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points. Let $D$ be a divisor satisfying the paper's condition; assume $D$ is effective. Using Laface's algorithm, which removes fixed $(-1)$-curve components and copies of the strict transform $\widetilde{C}_e$ of the negative section, call $D$ $(-1)$-special when the resulting divisor $M$ satisfies $v(M)>v(D)$. A divisor is special when its actual dimension exceeds its expected dimension. Laface's conjecture. $D$ is special if and only if it is $(-1)$-special. This is the Hirschowitz-type formulation of the SHGH conjecture in this setting; its validity is not established in general.