The non-injectivity conjecture for surface braid-group abelianization

Let $Br_\Delta$ be the braid group associated with a triangulation $\Delta$, and let

$$Br_\Delta^{ab}=Br_\Delta/[Br_\Delta,Br_\Delta]$$

be its abelianization. Abelianization non-injectivity conjecture. The canonical homomorphism

$$Br_\Delta\to Br_\Delta^{ab}$$

is never injective. Thus every braid group in this family has a nontrivial commutator subgroup. The preceding discussion establishes finite-order behavior for certain abelianized products, but does not prove this general non-injectivity assertion.