Characterizing symplectic capacities on ellipsoids

It is a long-standing conjecture that all symplectic capacities which are equal to the Gromov width for ellipsoids coincide on a class of convex domains in $\mathbb{R}^{2n}$. It is known that they coincide for monotone toric domains in all dimensions. In this paper, we study whether requiring a capacity to be equal to the $k^{th}$ Ekeland-Hofer capacity for all ellipsoids can characterize it on a class of domains. We prove that for $k=n=2$, this holds for convex toric domains, but not for all monotone toric domains. We also prove that for $k=n\ge 3$, this does not hold even for convex toric domains.