Existence of angle labelings for essentially 3-connected maps

Let $G$ be a map on a genus $g\geq 1$ orientable surface. An edge, positive-vertex, positive-face angle labeling means a labeling with the edge condition and positive vertex and face conditions indicated by those parameters. A map is essentially 3-connected when its universal-cover graph is 3-connected in the relevant essential sense. The angle-labeling conjecture. $G$ admits an edge, $\mathbb{N}^*$-vertex, $\mathbb{N}^*$-face angle labeling if and only if it is essentially 3-connected.

The “only if” direction is supported by the cited proposition and lemma; the converse is proposed as the general-map analogue of the preceding triangulation conjecture.