The subcontraction conjecture for facial 3-colorability and 3-flows

Let $G$ be a graph. A graph is facially $3$-colorable if it has a facial $3$-coloring, and $3$-flowable if it admits a nowhere-zero $3$-flow. For $n\geq 4$, let $K_{3,n}^+$ be the graph obtained from $K_{3,n}$ by joining two vertices in its partite set of size $3$ by an edge; a subcontraction is the graph operation used in the paper's definition.

The subcontraction conjecture. If $G$ is a facially $3$-colorable graph which does not have a subcontraction isomorphic to $K_{3,n}^+$ for some $n\geq 4$, then $G$ is $3$-flowable.

A positive answer would establish facial $3$-colorability and $3$-flowability as equivalent for graphs excluding the displayed obstructions as subcontractions. The paper proves this equivalence for graphs of maximum degree at most $3$ and for $K_{3,3}$-minor-free graphs, but the general assertion remains open.