Leung's weak pinching conjecture for minimal submanifolds

Let $W^n$ be a closed Riemannian manifold minimally immersed in the unit sphere $S^{n+p}(1)$. Let $B$ be its second fundamental form and define

$$\sigma(W)=\max\{|B(X,X)|^2\mid X\in TW,\ |X|=1\}.$$

Leung's weak pinching conjecture. If $n$ is odd and $\sigma(W)\leq\frac{n}{n-1}$, then $W$ is homeomorphic to $S^n$.

Leung proved the corresponding conclusion under the additional assumption that the normal connection is flat. The conjecture removes that assumption and concerns the topology forced by a pinching condition on the second fundamental form.