Bogomolov–Miyaoka–Yau inequality for four-manifolds with a Seiberg–Witten basic class

Let $X$ be a closed, connected, oriented, smooth four-dimensional manifold with $b^+(X)>1$ and odd $b^+(X)-b_1(X)$. A Seiberg–Witten basic class is a cohomology class associated with a $spin^c$ structure having nonzero Seiberg–Witten invariant. Bogomolov–Miyaoka–Yau conjecture. If $X$ has a Seiberg–Witten basic class, then

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

The conjecture is false when $b^+(X)=1$ and $c_2(X)<0$, so the stated hypotheses cannot be weakened in that direction.