Ozsváth–Szabó simple-type conjecture for symplectic 4-manifolds

Let $X$ be a symplectic $4$-manifold with $b^+(X)\geq 2$. The manifold $X$ has Ozsváth–Szabó simple type if every $\operatorname{spin}^c$ structure $\mathfrak{s}$ for which $\Phi_{X,\mathfrak{s}}\neq 0$ satisfies $d(\mathfrak{s})=0$. Ozsváth–Szabó simple-type conjecture. If $X$ is a symplectic $4$-manifold with $b^+(X)\geq 2$, then $X$ has Ozsváth–Szabó simple type. Taubes had shown that symplectic $4$-manifolds with $b^+\geq 2$ have Seiberg–Witten simple type, motivating this proposed Heegaard Floer analogue.