The slicing conjecture for convex bodies

For a convex body $\mathcal{K}\subset\mathbb{R}^{n}$, let $\operatorname{vol}_{n-1}(\mathcal{K}\cap H)$ denote the $(n-1)$-dimensional volume of its section by a hyperplane $H$. Define the slicing constant $L_n$ by

$$\frac{1}{L_n}:=\inf_{\mathcal{K}\subset\mathbb{R}^{n}}\sup_H\operatorname{vol}_{n-1}(\mathcal{K}\cap H),$$

where the infimum is over convex bodies of volume one and the supremum is over hyperplanes.

Slicing conjecture. The slicing constant is universally bounded:

$$L_n=O(1).$$

Following Guan's analysis, the source states that this conjecture was resolved by Klartag and Lehec and by Bizeul.