The $\ce1$-conjecture for Artin groups

Let $A_\Gamma$ be an Artin group and let $\chi:A_\Gamma\to\mathbb{R}$ be a character. The living subgraph $\mathrm{Liv}^\chi\subset\Gamma$ is obtained by deleting vertices $v$ with $\chi(v)=0$ and the interiors of edges $e=\lbrace u,v\rbrace$ with $\chi(u)+\chi(v)=0$ and even label $l(e)\geq 4$. Let $S(A_\Gamma)$ be the character sphere and let $\Sigma^1(A_\Gamma)\subseteq S(A_\Gamma)$. $\Sigma^1$-conjecture.

$$\Sigma^1(A_\Gamma)=\lbrace[\chi]\in S(A_\Gamma)\mid\mathrm{Liv}^\chi\text{ is connected and dominant}\rbrace.$$

This conjecture gives a combinatorial description of the first Bieri-Neumann-Strebel invariant for Artin groups; it is known for several families but remains open in general.