Conjecture I on the BNSR invariant of the weak commutativity quotient

Let $G$) be a group of type $FP_2$ (or finitely presented), and let $W(G)$ denote the subgroup used in the weak commutativity construction. The epimorphism $\mathfrak{X}(G)\to\mathfrak{X}(G)/W(G)$ identifies their character spheres.

Conjecture I. Under this identification,

$$\Sigma^2(\mathfrak{X}(G),\mathbb{Z})=\Sigma^2(\mathfrak{X}(G)/W(G),\mathbb{Z})$$

for groups of type $FP_2$, and

$$\Sigma^2(\mathfrak{X}(G))=\Sigma^2(\mathfrak{X}(G)/W(G))$$

when $G$ is finitely presented.

The conjecture follows from the cited theorems when $G'/G”$ is finitely generated, and holds for non-abelian limit groups. Equivalently, the inclusions in Corollary G(b) and (c) are equalities; the general case remains open.