The strong 4-coloring conjecture for graphs with one crossing

Let $G$ be a simple graph with a drawing in the plane containing one crossing, and call a triangle separating if deleting its vertices disconnects $G$. The vertices incident with the two crossed edges are the vertices incident with the crossing edges.

Strong one-crossing 4-coloring conjecture. If $G$ has no separating triangles and every vertex not incident with the crossed edges has degree at least five, then $G$ is 4-colorable.

This is presented as a stronger form of the minimum-degree conjecture for 5-critical graphs drawable with at most one crossing, and the source states that the exclusion of separating triangles is necessary; its status is not resolved in the supplied source.