The angle-labeling characterization of essentially 3-connected maps

Let $M$ be a map on a genus $g\geq 1$ orientable surface. An edge, positive-vertex, positive-face angle labeling is an angle labeling satisfying the edge, $\mathbb{N}^*$-vertex, and $\mathbb{N}^*$-face conditions. A map is essentially 3-connected in the sense used in the paper.

Angle-labeling characterization of essentially 3-connected maps. The map $M$ 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 established by a theorem in the source. The conjecture asks for the converse for general maps on orientable surfaces.