Negative curves conjecture for blow-ups of Hirzebruch surfaces

Let $\mathbb{F}_{e,r}$ denote the blow-up of the Hirzebruch surface $\mathbb{F}_e$ at $r$ very general points $p_1,\dots,p_r$. A curve is irreducible and reduced if it is an irreducible reduced curve on $\mathbb{F}_{e,r}$, and its self-intersection is negative when $C^2<0$. Negative curves conjecture. If $C$ is an irreducible and reduced curve in $\mathbb{F}_{e,r}$ with negative self-intersection, then $C$ is either a $(-1)$-curve or the strict transform of the negative section $C_e$ on $\mathbb{F}_{e,r}$. This is proposed as an analogue for Hirzebruch surfaces of the $(-1)$-curves conjecture for $\mathbb{P}^2$; the supplied text gives no resolution.