Sharp determinant conjecture at the canonical metric

Let $C$ be a smooth closed curve with canonical constant-curvature metric $g_{0}^{C}$, and let $l$ be a holomorphic line bundle with induced canonical metric $g_{0}^{l}$. For a Hermitian metric $g^{l}$ on $l$, let $A(g_{0}^{C},g^{l})$ denote the logarithm of the zeta-regularized determinant of the $\overline{\partial}$-Laplacian over $\mathcal{O}(l)$. Sharp determinant conjecture. $A(g_{0}^{C},g^{l})$ achieves its sharp upper bound only when $g^{l}$ is the standard metric $g_{0}^{l}$. This is the equality-case refinement of the determinant bound conjecture. The paper presents it as a conjecture motivated by the exact upper-bound problem; no resolution of this sharp equality statement is supplied here.