Viterbo's isoperimetric conjecture for symplectic capacities

Let $\Sigma \in \mathcal{K}^{2n}$ be a convex domain in $(\mathbb{R}^{2n},\omega_0)$, let $c$ be a symplectic capacity, and let $\operatorname{vol}(\Sigma)$ denote its symplectic volume. Viterbo's conjecture. For any symplectic capacity $c$ and any convex domain $\Sigma \in \mathcal{K}^{2n}$,

$$\frac{c(\Sigma)}{c(B^{2n})} \leq \left(\frac{\operatorname{vol}(\Sigma)}{\operatorname{vol}(B^{2n})}\right)^{1/n}.$$

This is an isoperimetric-type conjecture in symplectic geometry. It is known for convex bounded domains sufficiently near the symplectic ball, while the general case remains open.