Periodic Reeb flow and vanishing spectral gap equivalence conjecture

Let $(Y,\xi)$ be a closed contact manifold with contact form $\alpha$. Write $\mathfrak{gap}_\sigma(Y,\alpha)$ for the spectral gap associated with a class $\sigma\in CH(Y,\xi)$. Periodic spectral-gap equivalence conjecture. The following are equivalent: (a) the Reeb flow of $\alpha$ is periodic; and (b) there is a class $\sigma\in CH(Y,\xi)$ such that

$$\mathfrak{gap}_\sigma(Y,\alpha)=0.$$

The periodicity criterion proves the implication from (b) to (a), while this conjecture asserts the converse and is presented as a weaker reformulation of the preceding periodic spectral-gap conjecture. The converse remains open in the source.