Weak BMY Conjecture for symplectic 4-manifolds

Fix a finitely presented group $G$. Let $\mathfrak{M}(G)$ be the class of closed symplectic 4-manifolds with fundamental group $G$, and let

$$f_{\mathfrak{M}(G)}(b)=\inf_{M\in\mathfrak{M}(G)}\{\chi(M)+b\sigma(M)\}.$$

Define

$$D_{\mathfrak{M}(G)}=\{b\mid f_{\mathfrak{M}(G)}(b)\ne-\infty\},$$

and let $e_G$ be the left endpoint of $D_{\mathfrak{M}(G)}$.

Weak BMY Conjecture. For each finitely presented group $G$,

$$e_G=-3.$$

The BMY inequality $c_1^2\leq 9\chi_h$ is equivalent to $f_{\mathfrak{M}^{\min}(G)}(-3)\geq 0$ when $G$ is not a surface group. The conjecture weakens the BMY conjecture to a statement about the endpoint of the domain of the symplectic-manifold invariant $f_{\mathfrak{M}(G)}$; its resolution is not established in the supplied text.