The period-map surjectivity conjecture for four-manifolds

Let $X$ be a closed oriented $4$-manifold with $b_2^+(X)=1$. The period map sends a Riemannian metric to the self-dual line $H^+$ in the cone $\mathcal{P}$ of elements of positive square in $H^2(X;\mathbb{R})$, or equivalently to its projective class in $\mathbb{P}(\mathcal{P})$. Period-map surjectivity conjecture. The period map is surjective for all closed oriented $4$-manifolds with $b_2^+=1$. This conjecture was used by M. Katz in proving a theorem and remains open, including for blow-ups of $\mathbb{C}P^2$.