Infinitude of the étale fundamental group for non-normal-crossings section divisors

Determine whether the étale fundamental group of \(\mathbb{P}^{1}_{\mathbb{Z}}\setminus D\) is infinite whenever \(D\) is a reduced union of sections that is not a normal-crossings divisor at the prime \(2\).

Background

The paper discusses the étale fundamental group of the complement of a reduced union of sections in PZ1\mathbb{P}^{1}_{\mathbb{Z}}. An observation of T. Saito states that this fundamental group is trivial when the divisor is normal crossings. The cited work of Shimizu–Yamaguchi establishes that, if the divisor fails to be normal crossings at the prime $2$, the group has a quotient isomorphic to Z/2Z\mathbb{Z}/2\mathbb{Z}, but leaves unresolved whether the full étale fundamental group must consequently be infinite. These examples concern non-proper arithmetic surfaces whose generic fibers have genus zero.

References

They prove that this group admits a quotient isomorphic to Z/2Z whenever D is not normal crossings at 2, and ask whether it is then infinite Remark~3.8.

— A proper arithmetic surface with infinite étale fundamental group  (2610.05972 - Chenal, 5 Oct 2026) in Section 1, subsection “Some context” (discussion of Shimizu–Yamaguchi; citing Remark 3.8 of their work)