Harbourne–Janssen–Nagel–Schenck conjecture on negative curves on blowups of Hirzebruch surfaces

Let $\mathbb{F}_e$ be the $e$-th Hirzebruch surface, with $C_e$ denoting its negative section. Let $P=\{p_1,\dots,p_r\}\subset \mathbb{F}_e$ be $r$ points in very general position, and let

$$\pi_P:\mathbb{F}_{e,P}\to\mathbb{F}_e$$

be the blowup at the points in $P$. Harbourne–Janssen–Nagel–Schenck conjecture. If $C$ is an integral curve on $\mathbb{F}_{e,P}$ such that $C^2<0$, then $C$ is either the strict transform of $C_e$ or a $(-1)$-curve. This predicts that the negative curves on these blowups are exhausted by the distinguished negative section and exceptional-type curves; the supplied text gives no resolution status, so the conjecture is treated as open.