Canonical reduction for arithmetic modular heights

Let $(X,L)$ be a normal projective variety over a number field $K$, and consider integral projective models $(\mathcal{X},\mathcal{L})$ over rings of integers $\mathcal{O}_{K'}$ of finite extensions $K'/K$. Call a model $h_K$-minimising if it has minimal $h_K$ among all such models of $(X,L)$. Canonical reduction. An $h_K$-minimising model has reduced and semi-log-canonical geometric fibers; if the generic fiber is a klt $\mathbb{Q}$-Fano variety, every fiber is klt and $\mathbb{Q}$-Fano; and if $K_F\equiv a\mathcal{L}|_F$ with $a\geq0$, then $h_K(\mathcal{X},\mathcal{L})$ is minimal among all models if and only if every fiber $G$ is reduced and geometrically semi-log-canonical with $K_G\equiv a\mathcal{L}|_G$. This conjectural description is presented as the arithmetic counterpart of canonical or stable reduction, and the source gives no resolution evidence.