Nonalgebraic symplectic 4-manifolds on the Bogomolov–Miyaoka–Yau line

Determine whether there exists a symplectic 4-manifold on the Bogomolov–Miyaoka–Yau line that is not an algebraic surface.

Background

The reverse-engineering construction in Theorem 1.1 produces, under suitable assumptions on symplectic fillings, symplectic 4-manifolds lying on the Bogomolov–Miyaoka–Yau line. The paper notes that algebraic surfaces on this line are constrained by Yau’s theorem to be quotients of the complex 2-ball.

The unresolved issue is whether the symplectic category contains any examples on this line that do not arise from algebraic surfaces. This question concerns the gap between symplectic and complex-geometric realizations of the Bogomolov–Miyaoka–Yau equality.