Simon's quantization conjecture for closed minimal surfaces in spheres

Let $M$) be a closed surface minimally immersed in $\mathbb{S}^N$ such that its image is not contained in any hyperplane of $\mathbb{R}^{N+1}$. Set $S=|h|^2$, where $h$ is the second fundamental form, and define

$$S(s)=\frac{2(s-1)(s+2)}{s(s+1)}.$$

The Gauss equation gives $2K=2-S$, where $K$ is the Gaussian curvature. Simon's conjecture. If $s\in\mathbb{N}$ and

$$S(s)\leq S\leq S(s+1),$$

then either $S=S(s)$ or $S=S(s+1)$. Consequently, the immersion is one of Calabi's $2$-spheres, with ambient dimension $N=2s$ or $N=2s+2$, respectively. The first and second gaps are known, while the statement in this form concerns the broader quantization and gap problem for minimal surfaces in spheres.