The parity conjecture for cyclic triangulations of $S^2 \times S^1$

Let $M$ be a combinatorial $3$-manifold with a transitive cyclic automorphism group. Suppose that $M$ is homeomorphic to the total space of the orientable sphere bundle over the circle, $S^2 \times S^1$.

Parity conjecture. $M$ has an even number of vertices.

This conjecture arises from the classification of combinatorial $3$-manifolds with transitive cyclic symmetry. The source provides no resolution, so whether the parity assertion holds in general remains open.