Gillet–Soulé's determinant bound conjecture for line bundles on curves

Let $C$ be a compact complex curve with Hermitian metric $g^{C}$, and let $l$ be a line bundle over $C$ with associated Hermitian metric $g^{l}$. Write $A(g^{C},g^{l})$ for the logarithm of the zeta-regularized determinant of the $\overline{\partial}$-Laplacian over $\mathcal{O}(l)$. Gillet–Soulé's conjecture. $A(g^{C},g^{l})$ is bounded from above by a constant independent of the choices of $g^{C}$ and $g^{l}$. This conjecture is motivated by the Arithmetic Riemann–Roch theorem and has analytic interpretations involving spectral invariants, Toeplitz operators, and the Szegő limit theorem. It is proved in the paper for every holomorphic line bundle over $\mathbb{CP}^{1}$, but the general curve case is not resolved here.