The Xu–Gu Ricci-pinching sphere conjecture for submanifolds

Let $M^n$, with $n\geq4$, be a closed, simply connected, oriented submanifold of the space form $F^N(c)$. Let $Ric_M$ denote its Ricci curvature and $H$ its mean-curvature vector.

Xu–Gu's conjecture. If

$$Ric_M>(n-2)\left(c+\lvert H\rvert^2\right)>0,$$

then $M$ is diffeomorphic to $\mathrm{S}^n$.

The condition is motivated by rigidity and sphere theorems for submanifolds in space forms, including earlier results of Ejiri and Xu–Gu. The supplied text does not report a proof or disproof of this conjecture, so its status remains open here.