Marques–Neves linear Betti-number bound for minimal hypersurfaces

Let $(M,g)$ be an ambient manifold with positive Ricci curvature, and let $\Sigma$ be an embedded orientable compact minimal hypersurface of index $I$. Marques–Neves' Betti-number conjecture. The first Betti number of $\Sigma$ is bounded by a fixed multiple of $I$.

This conjecture predicts a linear relationship between the Morse index and topology for minimal hypersurfaces in positively Ricci-curved manifolds. The source states that it remains open; related index bounds are known in several settings, but no general linear bound of this form is given here.