Thomassen's conjecture for nonorientable surface triangulations

Let a triangulation of a nonorientable surface have Euler genus $g$, and let $h$ be an integer satisfying

$$1 \leq h < g.$$

Let an NSC be a noncontractible separating cycle; the two surfaces separated by it are understood after capping the boundary components with disks.

Nonorientable analogue of Thomassen's conjecture. Every such triangulation contains an NSC such that the two surfaces separated by the NSC have Euler genera $h$ and $g-h$, respectively.

This conjecture prescribes the Euler genera of the two pieces, strengthening the assertion that an NSC exists. The surrounding text presents it as an analogue of Thomassen's orientable-surface conjecture; no general resolution is given.