Coercivity conjecture for the analytic torsion functional

Let $X$ be a polarized manifold with reference Kähler form $\omega_{0}$, let $\mathcal{H}_{\omega_{0}}$ be the space of Kähler potentials, and let $\mathcal{F}_{\omega_{0}}$ be the corresponding functional. Assume that $\operatorname{Aut}_{0}(X)$ is trivial and that $\mathcal{F}_{\omega_{0}}$ admits a critical point in $\mathcal{H}_{\omega_{0}}$.

Analytic torsion coercivity conjecture. Under these assumptions, $\mathcal{F}_{\omega_{0}}$ is coercive.

The conjecture is motivated by the analogous principle for the anticanonical polarization: existence of a critical point should imply coercivity when there are no continuous automorphisms. The supplied text does not give a resolution.