The blow-up conjecture for ruled surfaces

Let $M$ be a ruled compact complex surface, and let $\widetilde{M}$ be its blow-up at sufficiently many points. Blow-up conjecture. The surface $\widetilde{M}$ admits a scalar-flat Kähler metric. This conjecture proposes that the genus hypothesis in the main theorem is unnecessary, including for ruled surfaces of genus $0$ and $1$.