Viterbo's volume-capacity conjecture for convex domains

Let $c$ be a symplectic capacity and let $X\subset\mathbb{R}^{2n}$ be a convex domain. Write $\operatorname{Vol}(X)$ for the Euclidean volume of $X$. Viterbo's conjecture. For every symplectic capacity $c$,

$$c^n(X)\leq n!\operatorname{Vol}(X).$$

Haim-Kislev and Ostrover provided a counterexample using the Hofer–Zehnder capacity for the Lagrangian product of a regular pentagon and its $90^\circ$ rotation, so the conjecture is false in general. Its restriction to Lagrangian products $K\times_L K^{\circ}$ of centrally symmetric convex bodies remains an open question and is connected to Mahler's conjecture.