Ganatra-Pomerleano's coordinate-function conjecture for cotangent bundles

Let $Q$ be a closed, oriented $3$-manifold. A coordinate function on a Liouville manifold $M$ is a pair $(c,\phi)\in\mathit{SH}^0(M)\times\mathit{SH}^1(M)$ satisfying

$$[c,\phi]=1.$$

Ganatra-Pomerleano's conjecture. If $T^*Q$ admits a coordinate function, then $Q$ is diffeomorphic to $S^1\times\Sigma_g$ for some $g\geq0$. The conjecture is presented as motivated by the three-dimensional theorem in the source, but no resolution is given.