Marques–Neves–Schoen index–Betti number conjecture

Let $(M,g)$ be a compact Riemannian manifold with positive Ricci curvature and dimension at least three. For a closed, embedded, orientable, minimal hypersurface $\Sigma\to M$, let $b_1(\Sigma)$ denote its first Betti number with real coefficients, and let $\operatorname{ind}(\Sigma)$ denote its Morse index.

Marques–Neves–Schoen conjecture. There exists $C>0$ such that, for all such hypersurfaces $\Sigma$, one has

$$\operatorname{ind}(\Sigma)\geq Cb_1(\Sigma).$$

This conjecture asks for a linear lower bound on the Morse index in terms of the first Betti number for minimal hypersurfaces in compact manifolds with positive Ricci curvature. The supplied source does not establish its resolution; its status is therefore open.