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

This is stated as a stronger conjecture motivated by the second fundamental form of Clifford minimal hypersurfaces in unit spheres. The source gives no resolution of the conjecture.