The arithmetic intersection conjecture for Gibbs partition functions

Let $(\mathcal{X},\mathcal{L})$ be an arithmetic variety as above, and let $X$ be its corresponding Fano manifold. Assume that $X$ admits a unique Kähler–Einstein metric, with volume form $dV_{KE}$ normalized to have unit total volume. The arithmetic intersection conjecture. As $k\to\infty$,

$$\frac{(n+1)!}{k^n}\log\mathcal{Z}_{N_k}$$

converges to the $(n+1)$-fold arithmetic self-intersection number of the line bundle $\mathcal{L}$, metrized by $dV_{KE}$. This conjecture connects the arithmetic partition functions of Fano models with Arakelov-theoretic intersection numbers; the source gives no resolution status.