The core conjecture for right-angled Artin groups

Let $G=\Gamma(\Gamma,\{\mathbb Z\})$ and $G'=\Gamma(\Gamma',\{\mathbb Z\})$ be right-angled Artin groups. Core conjecture. Then

$$G\equiv G'$$

if and only if the cores of $G$ and $G'$ are isomorphic. This proposes that the core is a complete invariant of the elementary theory of a right-angled Artin group.