Laface's modified Harbourne–Hirschowitz conjecture for Hirzebruch surfaces

Let ${\mathcal{L}}=|ah+bF-\sum_{i=1}^{h}m_iP_i|$ be a system on the Hirzebruch surface ${\mathbb{F}}_e$, and let ${\mathcal{M}}$ be its residual system after repeatedly subtracting any $(-1)$-curve having negative intersection and then subtracting the negative section $h$ whenever its intersection is negative. Laface's modified Harbourne–Hirschowitz conjecture. The system ${\mathcal{L}}(-\sum_{i=1}^hm_iP_i)$ is special if and only if it is $(-1)$-special, meaning $v({\mathcal{M}})>v({\mathcal{L}})$. This extends the proposed $(-1)$-curve characterization of speciality from the plane to Hirzebruch surfaces.