The core classification conjecture for right-angled Artin groups

Let $\mathbb G(\Gamma)$ and $\mathbb G(\Delta)$ be right-angled Artin groups, and let $\operatorname{Core}(\Gamma)$ denote the core graph. RAAG core classification conjecture. If

$$\operatorname{Core}(\Gamma)\simeq\operatorname{Core}(\Delta),$$

then $\mathbb G(\Gamma)\equiv\mathbb G(\Delta)$. Moreover, $\operatorname{Core}(\Gamma)$ is the RAAG with the smallest graph in its elementary class: if $\mathbb G(\Delta)\equiv\mathbb G(\operatorname{Core}(\Gamma))$, then $\operatorname{Core}(\Gamma)<\Delta$. If $\mathbb G(\Gamma)\equiv\mathbb G(\Delta)$, then $\Delta$ is obtained from $\operatorname{Core}(\Gamma)$ by adding redundant vertices. This would classify elementary equivalence of RAAGs by their cores; the first concrete consequence about non-abelian free groups is known by work of Sela and Kharlampovich–Miasnikov.