Anti-self-dual blowups

Let $X$ be a closed, oriented four-manifold containing an embedded sphere with self-intersection number $(-1)$. Suppose that $b_2^{+}(X)\le 3$. We show that there exists a Riemannian metric on $X$ such that the cohomology class dual to this sphere is represented by an anti-self-dual harmonic form. Furthermore, such a metric can be constructed even when there are multiple disjoint embedded $(-1)$-spheres.