The low-point Mori dream space conjecture for Hirzebruch-surface Hilbert schemes

Let $\mathbb{F}_r$ be the $r$th Hirzebruch surface, and let $\mathbb{F}_r^{[r-k]}$ denote its Hilbert scheme of $r-k$ points, where $r>k\geq 0$. Write $\overline{\operatorname{Eff}}(\mathbb{F}_r^{[r-k]})$ for its pseudo-effective cone.

Low-point Mori dream space conjecture. The Hilbert scheme $\mathbb{F}_r^{[r-k]}$ is a Mori dream space, and the decomposition of $\overline{\operatorname{Eff}}(\mathbb{F}_r^{[r-k]})$ is given by the walls defined in Theorem 4.15 of BC13.

This conjecture gives a partial converse to the preceding wall-inheritance proposition when the number of points is low compared with $r$. The cited wall description is intended to determine the full decomposition of the effective cone.