Liu–Xu–Ye–Zhao convergence conjecture for pinched submanifolds in spheres

Let $M_{0}$ be an $n$-dimensional complete submanifold in the sphere $\mathbb{S}^{n+q}\left(1/\sqrt{c}\right)$. Let $h$ be its second fundamental form, $H$ its mean curvature vector, and let $\alpha(n,|H|,c)$ denote the pinching function used in the source. Liu–Xu–Ye–Zhao's convergence conjecture. If

$$\sup_{M_{0}}\bigl(|h|^{2}-\alpha(n,|H|,c)\bigr)<0,$$

then the mean curvature flow with initial value $M_{0}$ converges to a round point in finite time, or converges to a totally geodesic sphere as $t\to\infty$. In particular, $M_{0}$ is diffeomorphic to $\mathbb{S}^{n}$. This conjecture concerns convergence under the optimal pinching condition in positive-curvature space forms; the source attributes it to Liu, Xu, Ye, and Zhao and gives no resolution beyond stating it as an open conjecture.