Kalai–Meshulam total Betti number 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 $G$, and let $\beta$ denote its total Betti number. For every induced subgraph $H$ of $G$, consider $\beta(\mathrm{Ind}(H))$.

Kalai–Meshulam total Betti number conjecture. For any graph $G$, we have

$$\beta(\mathrm{Ind}(H)) \leq 1$$

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 states that this conjecture remains open. It is a stronger topological analogue of the reduced Euler characteristic conjecture.