The Stein-degree boundedness conjecture for boundary components over characteristic-zero fields

Let $d\in \mathbb N$, let $t\in \mathbb R^{>0}$, and let $K$ be a field of characteristic zero. Let $(X,B)$ be a log Calabi–Yau pair over $K$ of dimension $d$, and let $S$ be a component of $B$ whose coefficient in $B$ is at least $t$. For the morphism $S\to \operatorname{Spec}K$, define its Stein degree by

$$\operatorname{sdeg}(S/\operatorname{Spec}K)=\dim_K H^0(S,\mathcal O_S).$$

Stein-degree boundedness conjecture over non-closed fields. The quantity

$$\operatorname{sdeg}(S/\operatorname{Spec}K)$$

is bounded from above depending only on $d$ and $t$.

This is the analogous conjecture over non-closed fields, motivated by arithmetic applications such as varieties over number fields. The source presents it as a formulation of interest rather than a resolved result.