The arithmetic Mabuchi infimum conjecture for log Fano and log general type pairs

Let $\mathbb{F}$ be a number field and let $(X_{\mathbb{F}},D_{\mathbb{F}})$ be a log pair such that $pm K_{(X_{\mathbb{F}},\Delta_{\mathbb{F}})}$ is ample. A polarized model $(\mathcal{X},\mathcal{D};\mathcal{L})$ ranges over all polarized models of $(X_{\mathbb{F}},\Delta_{\mathbb{F}};pm K_{(X_{\mathbb{F}},\Delta_{\mathbb{F}})})$ over $\mathcal{O}_{\mathbb{F}}$, while $(\mathcal{X},\mathcal{D})$ ranges over all models of $(X_{\mathbb{F}},\Delta_{\mathbb{F}})$ for which $pm\bra{\mathcal{K}}_{(\mathcal{X},\mathcal{D})}$ is relatively ample. Arithmetic Mabuchi infimum conjecture.

$$\inf_{(\mathcal{X},\mathcal{D};\mathcal{L})}\mathcal{\hat{M}}_{(\mathcal{X},\mathcal{D})}(\mathcal{L})=\inf_{(\mathcal{X},\mathcal{D})}\mathcal{\hat{M}}_{(\mathcal{X},\mathcal{D})}(\pm\mathcal{K}_{(\mathcal{X},\mathcal{D})}).$$

The conjecture says that the infimum of the arithmetic Mabuchi functional over all polarized models is achieved, at the level of the infimum, by the log-canonical polarization. It is posed in the context of relating canonical heights and arithmetic stability.