A note on the Chern Conjecture in dimension four

Let $M^4$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ and constant 3-mean curvature $H_3$ in ${\mathbb{S}}^5$. If $H_3^2\le \frac{1}{2}\mathrm{.}$ and Gauss-Kronecker curvature $K_M$ satisfies $K_M\le 1$ $($or $K_M\le \frac{S^2}{144}$$)$, then $M^4$ is isoparametric.