The planar graph braid-group Theta-four obstruction conjecture

Let $\Gamma$ be a planar graph, let $B_n\Gamma$ be its braid group, and let $P_3\Gamma$ be its pure braid group on three points. A group is simple-commutator-related if it has a presentation whose relators are commutators. Let $\Theta_4$ denote the four-edge theta graph. Theta-four obstruction conjecture. For $n\geq 3$, the groups $B_n\Gamma$ and $P_3\Gamma$ are simple-commutator-related if and only if $\Gamma$ does not contain a subgraph $\Theta_4$.

The conjecture is motivated by examples showing failure of the property for braid groups on graphs containing $\Theta_4$. The supplied text does not establish the converse or otherwise resolve the conjecture.