Boundedness of the maximal systolic constant for dynamically convex domains

Determine whether the maximal systolic constant Sys(DC^{2n}), defined as the supremum over dynamically convex domains in R^{2n} of the minimal action among closed characteristics on the boundary divided by (n! Vol)^{1/n}, is bounded above by a finite constant.

Background

The class DC^{2n} of dynamically convex domains generalizes convexity in symplectic dynamics. For this class, the authors define an analogue of the maximal systolic constant by replacing the Ekeland–Hofer–Zehnder capacity with the minimal action among closed characteristics on the boundary.

It is known that Sys(DC^4) ≥ 2, but the existence of a universal upper bound for Sys(DC^{2n}) in any dimension remains unresolved.