The symplectic Bogomolov–Miyaoka–Yau conjecture

Let $(X,\omega)$ be a closed, minimal symplectic $4$-manifold satisfying

$$c_1(X)^2 \geq 0.$$

Symplectic Bogomolov–Miyaoka–Yau conjecture. One has

$$c_1(X)^2 \leq 3c_2(X).$$

This is the proposed extension of the Bogomolov–Miyaoka–Yau inequality from complex surfaces of general type to closed symplectic $4$-manifolds. The source presents the inequality as conjectural and gives no resolution.