Arithmetic Mabuchi period conjecture for K-polystable pairs

Let $(\mathcal{X},\mathcal{D})$ be an arithmetic log pair over $\mathbb{Z}$ such that $-\mathcal{K}_{(\mathcal{X},\mathcal{D})}$ is a relatively ample $\mathbb{Q}$-line bundle, and suppose that its complexification $(X,\Delta)$ is K-polystable with quantized Futaki character vanishing for sufficiently large $k$. Let $\mathcal{Z}_N(\mathcal{X},\mathcal{D})$ be the canonical arithmetic period and let $\mathcal{M}_{\mathbb{Z}}$ be the arithmetic Mabuchi functional. Arithmetic Mabuchi period conjecture.

$$-\lim_{N\to\infty}N^{-1}\log\mathcal{Z}_N(\mathcal{X},\mathcal{D})=\inf\mathcal{M}_{\mathbb{Z}}.$$

This generalizes the cited arithmetic K-polystability conjecture to nontrivial symmetry groups and connects arithmetic periods with the infimum of the arithmetic Mabuchi functional. The source provides no resolution.