Ganatra-Pomerleano's cyclic coordinate-function conjecture

Let $Q$ be a closed, oriented $3$-manifold. A cyclic coordinate function on a Liouville manifold $M$ is a pair $(\tilde{c},\tilde{\phi})\in\mathit{SH}_{S^1}^1(M)\times\mathit{SH}_{S^1}^1(M)$ satisfying

$$\{\tilde{c},\tilde{\phi}\}=1.$$

Cyclic coordinate-function conjecture. If $T^*Q$ admits a cyclic coordinate function, then $Q$ is diffeomorphic to $S^1\times\Sigma_g$, where $g\geq0$, or a spherical space form. The source gives evidence for this expectation via the analogous coordinate-function conjecture, but no resolution is given.