Existence of a minimal punctured Klein bottle filling

Determine whether a minimal embedded punctured Klein bottle with boundary a great circle exists; equivalently, resolve the stated claim that no such surface exists.

Background

The paper discusses which genera and Euler numbers are topologically admissible for smooth embedded non-orientable surfaces bounded by an unknot. A punctured Klein bottle is the smallest relevant topology beyond the Möbius band, and the authors note that any counterexample to the proposed nonexistence statement would have Euler number ±4 or 0.

References

It is natural to ask which other simple topological types and Euler numbers can occur: There is no minimal embedded punctured Klein bottle with boundary a great circle.

— Minimal equatorial fillings  (2609.26701 - Bernstein et al., 22 Sep 2026) in Section 7, Question \ref{punctured}