The converse criterion for $\ce2$-invariants of Artin groups

Let $A_\Gamma$ be an Artin group satisfying the $K(\pi,1)$-conjecture, and let $\chi:A_\Gamma\to\mathbb{R}$ be a character. Write $\mathrm{Liv}^\chi$ for the living subgraph, and call a vertex or edge dead as defined in the paper. Let $\mathrm{slk}_{\mathrm{Liv}^\chi,v}^\chi$ denote the spherical link used in the criterion, and let the associated simplicial complex be obtained from $\mathrm{Liv}^\chi$ by attaching $2$-cells along all spherical triangles except those of type $\mathbb{B}_3$ with a $3$-dead edge. Converse $\ce2$ criterion. Then $[\chi]\in\Sigma^2(A_\Gamma,\mathbb{Z})$ (respectively, $[\chi]\in\Sigma^2(A_\Gamma)$) if and only if the following conditions hold: for every edge $e=\lbrace v,w\rbrace$ for which either $\chi(v)=\chi(w)=0$ or $e$ is dead, there is a living vertex $u$ such that $A_{\{v,w,u\}}$ is spherical; for every dead vertex $v$, $\mathrm{slk}_{\mathrm{Liv}^\chi,v}^\chi$ is non-empty and connected; and the resulting simplicial complex is $1$-acyclic (respectively, simply connected). This conjecture seeks necessity as well as sufficiency for a tractable characterization of the second BNSR invariants. The stated conditions are sufficient by Theorem 1.2, while the converse is presented as an open conjecture.