Finiteness conjecture for refined Hofer–Zehnder capacities of inessential manifolds

Let $M$ be an inessential closed compact manifold. Write $\langle \pi_1(M)\rangle$ for the set of conjugacy classes of $\pi_1(M)$, and let $\Gamma\subset \langle \pi_1(M)\rangle$ be a non-empty subset. The refined Hofer–Zehnder capacity $C_{HZ}^{\Gamma}(DM)$ is defined using Hamiltonians whose non-constant periodic orbits in the conjugacy classes in $\Gamma$ have period greater than $1$.

Finiteness conjecture. For every such $\Gamma$, the capacity $C_{HZ}^{\Gamma}(DM)$ is finite.

The source presents this as a conjecture about refined symplectic capacities of unit disc bundles; its resolution is not specified in the supplied material.