Finiteness for minimal properly elliptic surfaces over the projective line

Prove or disprove that the set of isomorphism classes of log canonical model targets is finite for every smooth projective minimal properly elliptic surface over \(\mathbb{P}^1\).

Background

The paper establishes finiteness of log canonical models for smooth projective minimal surfaces in every case except possibly minimal properly elliptic surfaces over P1\mathbb{P}^1. It supplies additional evidence for finiteness by proving the result when such a surface has a section and all fibers are irreducible, but leaves the general case unresolved.

References

Hence the only remaining case in which the finiteness of $\CC(X)$ is unknown is that of a minimal properly elliptic surface over $\Pp1$.

— Log canonical models of a fixed variety with varying boundaries  (2609.03305 - Li et al., 3 Sep 2026) in Proof of Theorem 2.7, Section 2.5; see also the question preceding the theorem in the Introduction