Li's conjecture on the threshold for positive curvature operator of the second kind

Let $M^n$ be a closed $n$-dimensional Riemannian manifold, and let $n+\frac{n-2}{n}$ be the positivity threshold for its curvature operator of the second kind. Li's conjecture. If $M$ has $(n+\frac{n-2}{n})$-positive curvature operator of the second kind, then $M$ is diffeomorphic to a spherical space form. This proposed sphere theorem concerns the sharp-looking positivity threshold suggested by the paper's results; the source provides no resolution, so the conjecture remains open.