Hamiltonian Seifert conjecture for prescribed aperiodic energy values

Let $H_0$ be the proper Hamiltonian on ${\mathbb R}^{2n}$ under discussion, let $K\subset(0,\infty)$ be a compact set of measure zero, and let $H\colon {\mathbb R}^{2n}\to {\mathbb R}$ be a Hamiltonian. Hamiltonian Seifert conjecture. For $2n\geq 6$, there exists a $C^\infty$-function $H\colon{\mathbb R}^{2n}\to {\mathbb R}$, $C^0$-close and isotopic with compact support to $H_0$, such that the Hamiltonian flow of $H$ has no closed trajectories for energy values in $K$. This would show that the set of aperiodic energy values can be an arbitrary prescribed compact measure-zero set, extending known constructions of Hamiltonians with aperiodic levels.