Admissibility conjecture for wiring diagrams without oriented faces

Let $G$ be a wiring diagram on a torus. A face of $G$ is oriented if its boundary is oriented. The diagram $G$ is called admissible when it satisfies the admissibility condition used for the q-6V model.

Admissibility conjecture. If $G$ includes no oriented faces, then it is admissible.

This conjecture gives a sufficient combinatorial condition for admissibility, which is needed for the commutativity of the transfer matrices. The supplied text does not state whether the conjecture has been proved or disproved.