Source-vertex broken hexagon conjecture for modified crystal operator components

Let $C$ be a connected component of the graph given by the modified operators $\phi_i$, and call a hexagon broken by $\phi_i$ after $\phi_j$ when it has the broken-hexagon property described in the preceding remark. Source-vertex broken hexagon conjecture. If $C$ contains a broken hexagon broken by some $\phi_i$ after some $\phi_j$, then the source vertex of $C$ also has a broken hexagon broken by $\phi_i$ after some $\phi_j$. The claim concerns the structure of connected components of the modified-operator graph; the source says it has been confirmed by computer testing, but supplies no proof, so it remains open.