Conjecture on normalized symplectic and real symplectic capacities

Let $D$ be a convex domain in $\mathbb{R}^{2n}$ invariant under a fixed linear anti-symplectic involution. Capacity coincidence conjecture. All normalized symplectic capacities and normalized real symplectic capacities of $D$ are the same. This conjecture is motivated by the classical question of whether all normalized symplectic capacities agree on compact convex domains; the provided text gives no resolution status for the real-symplectic version.