The isotropic constant conjecture

For a convex body $K\subseteq\mathbb{R}^n$, let $L_K$ denote its isotropic constant. The isotropic constant conjecture. There is a universal constant $c>0$ such that, for every $n\geq2$ and every convex body $K\subseteq\mathbb{R}^n$,

$$L_K\leq c.$$

The paper states that this formulation is equivalent to Bourgain's hyperplane conjecture and explicitly says that it remains unresolved, although increasingly strong dimension-dependent bounds are known.