Existence of analytic torsion critical points on curves

Let $L\to X$ be an ample line bundle over a complex curve $X$ of genus at least one, and let $\mathcal{F}_{k}$ be the associated analytic torsion functionals.

Curve critical-point conjecture. For every $k$, the functional $\mathcal{F}_{k}$ admits a critical point.

The conjecture is presented as a more precise form of the preceding existence statement, motivated by the limiting constant-curvature problem. In genus one, the source says that it would, together with the paper's results, confirm the Gillet–Soulé conjecture.