The Willmore functional lower-bound conjecture for genus-zero hypersurfaces

Let $M^n$ be a compact hypersurface of genus $0$ in $\mathbb{R}^{n+1}$ whose volume equals that of the unit sphere, and let $\mathcal{W}(M)$ denote its Willmore functional. Willmore lower-bound conjecture. Every such hypersurface satisfies

$$\mathcal{W}(M)\geq n.$$

Moreover,

$$\mathcal{W}(M)=n$$

if and only if $M$ is the Euclidean sphere. The conjecture extends the local maximality result established for the unit sphere and proposes a global characterization of the equality case among compact genus-zero hypersurfaces with the same volume.