Alpert–Kahle–MacPherson homological stability conjecture for sliding fifteen-puzzle configurations

Let $R_{w,h}$ be a rectangle of width $w$ and height $h$, let $SF_n(R_{w,h})$ denote the space of configurations of $n$ tiles in the rectangle with sliding fifteen-puzzle moves, and let $F_n(\mathbb{R}^2)$ be the configuration space of $n$ points in the plane. Assume

$$w\ge h\ge k+2,\qquad wh-n\ge (k+1)(k+2).$$

Alpert–Kahle–MacPherson's conjecture. The inclusion of $SF_n(R_{w,h})$ into $F_n(\mathbb{R}^2)$ induces an isomorphism

$$H_k\big(SF_n(R_{w,h})\big)\cong H_k\big(F_n(\mathbb{R}^2)\big).$$

The conjecture was positively resolved in the source for general $k$ when $h=k+2$ or $k+3$, and for $k=1$ under the sharper empty-space condition $wh-n\ge 6$.