The minimum $b_2$ conjecture for right-angled Artin groups

Let $G$ be a right-angled Artin group, and let $h(G)$ be the minimum of $b_2(M)$ over all closed, oriented topological $4$-manifolds $M$ with fundamental group isomorphic to $G$. Let $b_2(G)$ denote the second Betti number of $G$, and let $m_2(G)$ be the maximum rank of the symmetric bilinear form

$$H^2(G;\mathbb{Z}_2)\times H^2(G;\mathbb{Z}_2)\longrightarrow\mathbb{Z}_2, \qquad (a,b)\longmapsto (a\cup b)\cap\alpha$$

over all choices of $\alpha\in H_4(G;\mathbb{Z}_2)$. Minimum $b_2$ conjecture. For every right-angled Artin group $G$,

$$h(G)=2b_2(G)-m_2(G).$$

The lower bound is known for all finitely presented groups, and equality has been established for several infinite families of right-angled Artin groups. The conjecture asserts that this lower bound is always sharp for right-angled Artin groups, extending the exact calculations known for free and free abelian groups; the general case remains open.