Parity of oriented vertices and surface boundaries

Prove that, if an oriented 6-cycle 4-cover splits into two 6-cycle double covers or has all edges rich, then the number of oriented vertices has the same parity as the number of boundary components of the associated 6-cycle 4-cover surface.

Background

The paper constructs orientable surfaces from certain oriented 6-cycle 4-covers and observes a parity relationship between local vertex types and the resulting boundary components. The relationship is stated for covers that split into two 6-cycle double covers or whose edges are all rich; the all-disordered case is noted to be immediate, but the general claim is left as a conjecture.