The Mori-surface conjecture for Hirzebruch surfaces

Let $\mathbb{F}_r$ denote the $r$th Hirzebruch surface.

Mori-surface conjecture. The surface $\mathbb{F}_r$ is a Mori surface.

This conjecture is introduced as a further statement that would imply that the word “linear” is unnecessary in the preceding proposition. Its status is not resolved in the supplied text.