Homotopy type conjecture for higher independence complexes of rectangular grid graphs

For $m,n\geq 2$, let $G_{m,n}$ be the rectangular grid graph with vertex set

$$V(G_{m,n})=\{(i,j):i\in[m],\ j\in[n]\}$$

and edges joining $(i,j)$ to $(i+1,j)$ or $(i,j+1)$ whenever those vertices exist. For an integer $r$, let $\operatorname{Ind}_r(G_{2,n})$ denote the $r$-independence complex of the $2\times n$ grid graph. Grid-graph homotopy conjecture. For all $r\geq n$, $\operatorname{Ind}_r(G_{2,n})$ is either contractible or homotopy equivalent to a wedge of spheres of dimension $r-1$. This is based on computations of the complexes and their homology for small values of $n$. The conjecture concerns the homotopy types of higher independence complexes of grid graphs, extending the study of ordinary independence complexes in this setting; no general proof is given.