Dual-acyclic orientation conjecture for positive-genus triangulations

Let $G$ be a triangulation on a genus $g\geq 1$ orientable surface. An orientation of $G$ has vertex outdegree divisible by $3$ and at least $3$ when every vertex has outdegree divisible by $3$ and at least $3$; the dual orientation is obtained by orienting dual edges consistently with the primal orientation. The dual-acyclic orientation conjecture. $G$ admits an orientation of its edges such that every vertex has outdegree at least $3$, divisible by $3$, and such that there is no oriented non-contractible cycle in the dual orientation.

Such an orientation would generalize the cited positive-genus orientation theorem in a form suitable for preserving the genus of the associated spanning unicellular map; its existence is presented as an open higher-genus question.