Boundedness of the maximal systolic constant for dynamically convex domains

Ascertain whether the maximal systolic constant Sys(DC^{2n}) for the class DC^{2n} of dynamically convex domains in R^{2n}, defined analogously to the convex case by replacing the Ekeland–Hofer–Zehnder capacity with the minimal action among closed characteristics on the boundary, is bounded above.

Background

Dynamically convex domains form a class that includes convex domains; recent results show dynamically convex domains need not be symplectically convex. The maximal systolic constant Sys(DC^{2n}) is defined by normalizing the minimal action among closed characteristics by ((n! Vol)^{1/n}), analogously to the convex setting.

A lower bound Sys(DC^{4}) ≥ 2 is known, but whether Sys(DC^{2n}) is bounded above for any (or all) dimensions remains unknown.