Almost nonnegative curvature operator Betti-number conjecture

Let $m$ be a positive integer, and let $(M^m,g)$ be a compact Riemannian manifold with diameter $\operatorname{diam}(M)=1$. Write $\mathcal{R}m$ for its curvature operator, and let $\text{b}_i(M)$ denote its $i$-th Betti number. Almost nonnegative curvature operator Betti-number conjecture. There exists an $\epsilon(m)>0$ such that, if

$$\mathcal{R}m\ge -\epsilon g\odot g,$$

then

$$\text{b}_i(M)\le \binom{m}{i}.$$

Moreover, equality holds if and only if $M$ is diffeomorphic to a torus. The conjecture proposes a stability version of the equality statement for compact manifolds with nonnegative curvature operator; the source gives no resolution, so its status remains open.