The presentation conjecture for the pure braid group of points in the affine plane

Let $P_n^2$ be the pure braid group of motions of $n$ points in ${\Bbb A}^2$ such that no three points become collinear, and let $PL_n$ be the group defined by the presentation in the paper. There is a surjective homomorphism

$$\varphi_n:PL_n\to P_n^2.$$

Presentation conjecture. The homomorphism $\varphi_n:PL_n\to P_n^2$ is an isomorphism.

The conjecture proposes that the stated presentation completely describes the pure braid group of points in the affine plane. The paper establishes that $\varphi_n$ induces isomorphisms on the first and second integral homology groups, but does not establish that the homomorphism itself is an isomorphism.