Viterbo's isocapacitary conjecture for convex bodies

Let $K,Q e othing$ be convex bodies in $\mathbb{R}^n$, and let $c(K\times Q)$ denote the symplectic capacity of their Lagrangian product. Viterbo's isocapacitary conjecture. The Lagrangian product satisfies

$$\operatorname{vol}(K\times Q)\geq\frac{c(K\times Q)^n}{n!}.$$

This is a natural isoperimetric inequality in symplectic geometry and is related to the problem of finding smallest $Q$-covers. The paper discusses counterexamples to the conjecture in its greatest generality, so the stated general claim is refuted.