Periodic spectral-gap conjecture for contact homology

Let $(Y,\xi)$ be a closed contact manifold with a periodic contact form $\alpha$ of period $T$. Let $CH(Y,\xi)$ denote contact homology, $\mathfrak{s}_\sigma(Y,\alpha)$ the spectral invariant of a class $\sigma$, and $U_P$ a $U$-map. Periodic spectral-gap conjecture. There exists a class $\sigma\in CH(Y,\xi)$ and a $U$-map $U_P$ such that

$$\mathfrak{s}_{U_P\sigma}(Y,\alpha)=\mathfrak{s}_\sigma(Y,\alpha)=T.$$

This conjecture generalizes the paper's periodic spectral-gap theorem from the special setting considered there to arbitrary periodic contact forms; the source suggests a proof using Morse functions on the orbifold of closed Reeb orbits, intersection theory, and automatic transversality.