Generalized Chern-Hamilton conjecture for closed contact 3-manifolds

Let $(M,\zeta)$ be a closed contact $3$-manifold, where $\zeta$ is a contact distribution, and consider the Chern-Hamilton energy functional on the space of compatible metrics for $(M,\zeta)$. Generalized Chern-Hamilton conjecture. There exists a compatible metric that realizes the minimum of the Chern-Hamilton energy functional among all compatible metrics. This conjecture extends the original Chern-Hamilton conjecture by allowing the contact form to vary while fixing the contact distribution. The original conjecture for contact forms whose Reeb vector fields induce Seifert foliations was solved affirmatively, but the generalized existence assertion remains open.