Conjecture on nonseparating cycle types in nonorientable triangulations

Let $N_g$ denote the nonorientable surface of Euler genus $g$. A nonseparating cycle is one-sided when its neighborhood is a Möbius band, two-sided otherwise; it is orientable-leaving when deleting it leaves an orientable surface, and nonorientable-leaving otherwise.

Conjecture on nonseparating cycle types. If $g \geq 3$ is odd, every triangulation of $N_g$ has a nonseparating cycle that is one-sided and orientable-leaving, one that is one-sided and nonorientable-leaving, and one that is two-sided and nonorientable-leaving. If $g \geq 4$ is even, every triangulation of $N_g$ has a nonseparating cycle that is one-sided and nonorientable-leaving, one that is two-sided and orientable-leaving, and one that is two-sided and nonorientable-leaving.

The assertion extends the verified low-genus pattern for $N_1$, $N_2$, $N_3$, and $N_4$. No proof or counterexample is supplied for the stated general ranges.