The thin-shell conjecture for isotropic logconcave measures

For an isotropic logconcave probability measure $\pi$ on $\mathbb{R}^{n}$, define the thin-shell constant by

$$\sigma_{\mathsf{TS}}^{2}(n)=\sup_{\text{isotropic }\pi}\frac{1}{n}\operatorname{Var}_{\pi}(\|\cdot\|^{2}).$$

Thin-shell conjecture. The thin-shell constant is universally bounded:

$$\sigma_{n}^{2}=O(1).$$

The source describes this conjecture as resolved in July 2025, so its database status is recorded as solved.