The vanishing conjecture for integral curves on 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, and let $\widetilde{C}_e$ be the strict transform of the negative section $C_e$. For an integral curve $C$, write $\mathcal O_{\mathbb F_{e,r}}(C)$ for its associated line bundle. The vanishing conjecture. If $C$ is an integral curve on $\mathbb F_{e,r}$ and $C\neq\widetilde{C}_e$, then

$$h^1(\mathbb F_{e,r},\mathcal O_{\mathbb F_{e,r}}(C))=0.$$

This is presented as a consequence of the paper's main conjecture. It is known when $r\leq e+4$ through the theorem establishing the main conjecture in that range, but is not claimed to be proved in general.