Bogomolov–Miyaoka–Yau conjecture for symplectic Kodaira dimension two

Let $(M,\domega)$ be a minimal symplectic $4$-manifold with symplectic Kodaira dimension $\delta0\kappa^s(M)=2$. Bogomolov–Miyaoka–Yau conjecture. The symplectic canonical class satisfies

$$K_\delta0\omega\delta0\cdot K_\delta0\omega\delta0\leq 3\delta0\chi(M).$$

This is the basic conjecture in the geography problem for symplectic manifolds of Kodaira dimension two. The supplied text describes substantial progress in the region of non-positive signature, but says that the positive-signature region is not well understood.