Gillet–Soulé's non-negativity conjecture for arithmetic-surface cohomology

Let $\pi: X\rightarrow B=\operatorname{Spec}(\mathcal{O}_{K})$ be an integral, flat, projective scheme of dimension $2$, and let $L$ be a metrized line bundle on $X$. Suppose there is a reasonable cohomology theory associated to $X$, with first cohomological quantity $h^{1}(X,L)$. The Dolbeault Laplacian associated to the metrized line bundle is denoted by $\overline{\partial}^{*}\overline{\partial}$. Gillet–Soulé's conjecture. The quantity $h^{1}(X,L)$ is non-negative, and $h^{1}$ should be related to

$$\det(\Delta_{\overline{\partial}^{*}\overline{\partial}}).$$

The construction of a cohomology theory for arithmetic surfaces in Arakelov theory remains open; in particular, the proposed non-negativity is not known. The relationship with the determinant of the Dolbeault Laplacian is also left informal in the source.