The symplectic Poincaré-type uniqueness conjecture for minimizers

Let $G$ be a group, and let $M$ be a symplectic 4-manifold with $\pi_1(M)\cong G$ that minimizes $\chi:\mathfrak{M}(G)\rightarrow {\mathbb Z}$. Let $Q_M$ denote the intersection form of $M$. Symplectic minimizer uniqueness conjecture. Any other symplectic manifold with intersection form $Q_M$ that also minimizes $\chi:\mathfrak{M}(G)\rightarrow {\mathbb Z}$ is diffeomorphic to $M$.

The conjecture refines uniqueness of symplectic Euler-characteristic minimizers by fixing the intersection form, motivated by the existence of distinct minimizers with the same fundamental group but different intersection forms. The source describes the question as a conjecture, and the supplied status evidence does not establish a counterexample; its resolution should therefore be checked.