The Stein-degree boundedness conjecture for horizontal boundary components

Let $d e \in \mathbb N$ and let $t\in \mathbb R^{>0}$. A \log Calabi–Yau fibration $(X,B)\to Z$ of dimension $d$ consists of an lc pair $(X,B)$ and a contraction $X\to Z$ such that

$$K_X+B\sim_\mathbb R 0/Z.$$

For a projective morphism $S\to Z$, its Stein degree is

$$\operatorname{sdeg}(S/Z):=\deg(V/Z),$$

where $S\to V\to Z$ is the Stein factorisation, and it is $0$ if $S\to Z$ is not surjective. Let $S$ be a component of $B$ horizontal over $Z$, with coefficient in $B$ at least $t$.

Stein-degree boundedness conjecture. The quantity $\operatorname{sdeg}(S/Z)$ is bounded from above depending only on $d$ and $t$.

The preceding theorem establishes the analogous bound for every non-klt centre, and the conjecture gives a more general form for horizontal divisorial components. The source also suggests extensions to generalised pairs, non-horizontal components, slc fibrations, and non-closed fields; their status is not resolved here.