Structural reformulation of the 3-colorability conjecture for ${\cal G}_2$

Let ${\cal G}_{\ell}$ be the family of graphs with girth $2\ell+1$ and with no odd holes of length at least $2\ell+3$. Let $\theta^+$ be the graph obtained from the Petersen graph by removing two adjacent vertices, let $\theta$ be obtained from $\theta^+$ by removing an edge incident with two 3-vertices, and let $\theta^-$ be obtained from the Petersen graph by removing three vertices that induce a path.

Structural reformulation. Graphs in ${\cal G}_0$ induce neither $\theta$ nor $\theta^-$.

This is presented as an equivalent reformulation of the earlier 3-colorability conjecture after reducing graphs in ${\cal G}_2$ that contain two edge-sharing $5$-cycles to the configurations $\theta$ and $\theta^-$. The supplied material does not state whether this reformulation has been resolved.