Conjecture on constant-mean-curvature hypersurfaces in the four-sphere

Let $M$ be a closed hypersurface with constant mean curvature $H$ in $\mathbb{S}^{4}$, and let $S$ denote the squared length of its second fundamental form. Write $\beta(3,H)$ for the quantity used in the paper. The four-sphere constant-mean-curvature conjecture. (i) If $\beta(3,H)\leq S\leq 6+9H^2$, then $S\equiv\beta(3,H)$ or $S\equiv 6+9H^2$; equivalently, $M$ is a Clifford torus or a tube of the Veronese surface. (ii) If $S\geq 6+9H^2$, then $S\equiv 6+9H^2$, so $M$ is a tube of the Veronese surface. The paper explicitly notes that the case $H=0$ remains open, while the constant-scalar-curvature case was known under additional assumptions.