The planar graph braid group commutator presentation conjecture

Let $\Gamma$ be a planar graph, and let $B_2\Gamma$ denote its braid group on two strands. A presentation for $B_2\Gamma$ should exist in which every relator is a commutator associated to a pair of disjoint simple loops.

Planar graph braid group commutator presentation conjecture. There exists a presentation for $B_2\Gamma$ such that the relators are all commutators corresponding to pairs of disjoint simple loops.

This would generalize the preceding theorem, which establishes the analogous presentation result in the setting treated there. The parser supplies no evidence that the conjecture has been resolved.