The wedge-of-spheres conjecture for path-free complexes of complete multipartite graphs

Let $G=K_{n_1,\ldots,n_r}$ be a complete multipartite graph, and let $t\geq 2$. The path-free complex $\operatorname{PF}_t(G)$ is the simplicial complex whose faces avoid paths of length $t$ in $G$.

Wedge-of-spheres conjecture. The complex $\operatorname{PF}_t(G)$ is homotopy equivalent to a wedge of spheres.

The preceding cases establish this homotopy type for the values of $t$ treated in the paper, and the conjecture proposes the same conclusion for every $t\geq 2$. Determining the number and dimensions of the spheres in general remains open.