Gillet–Soulé's boundedness conjecture for Dolbeault determinants

Let $(X,\omega_{0})$ be a complex curve with a fixed Hermitian metric, and let $L\to X$ be a holomorphic line bundle. For a smooth Hermitian metric on $L$, let $\Delta_{\bar{\partial}}$ denote the Dolbeault Laplacian.

Gillet–Soulé's conjecture. The determinant of $\Delta_{\bar{\partial}}$, viewed as a functional on the space of all smooth Hermitian metrics on $L$, is bounded from above.

The conjecture is motivated by Arakelov geometry, in particular the arithmetic Riemann–Roch theorem, and is equivalent to boundedness from below of certain arithmetic Betti numbers. It was confirmed for $S^{2}$ by Fang.