Leung's spherical minimal-submanifold homeomorphism conjecture

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

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

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

Leung's conjecture strengthens the known rigidity result under a flat normal connection, which implies that such a submanifold is totally geodesic. The general homeomorphism assertion is presented as a conjecture in the source.