Viterbo's symplectic isoperimetric conjecture

Let $(\mathbb{R}^{2n},\omega_0)$ be standard symplectic space, let $c$ be a symplectic capacity, and let $\Sigma\subset(\mathbb{R}^{2n},\omega_0)$ be a convex domain. Write $\operatorname{vol}$ for symplectic volume.

Viterbo's conjecture.

$$\frac{c(\Sigma)}{c(B_2^{2n})}\leq\left(\frac{\operatorname{vol}(\Sigma)}{\operatorname{vol}(B_2^{2n})}\right)^{1/n}.$$

The conjecture is an isoperimetric inequality for symplectic capacities, asserting that the Euclidean ball optimizes the capacity-to-volume relation among convex domains. The source describes it as open, while noting that a special six-dimensional case was solved positively.