Engström's sphere-or-contractible conjecture for independence complexes

Let $G$ be a graph. Its independence complex $\mathrm{Ind}(G)$ is the simplicial complex whose faces are the independent sets of $G$. For every induced subgraph $H$ of $G$, consider the homotopy type of $\mathrm{Ind}(H)$.

Engström's sphere-or-contractible conjecture. For any graph $G$, we have that $\mathrm{Ind}(H)$ is contractible or homotopy equivalent to a sphere for every induced subgraph $H$ of $G$ if and only if there are no induced cycles of length divisible by three in $G$.

The source proposes this as an amendment to the Kalai–Meshulam sequence. The corresponding assertion is proved in the paper when $G$ has no cycles of length divisible by three, but the general induced-cycle characterization remains open.