The quasi-stability conjecture for parabolic bundles on ruled surfaces

Let $M$ be a blown-up ruled surface of genus at least $2$, and let $[\omega]$ be an admissible Kähler class, meaning that $c_1\cup[\omega]=0$. Associate to $[\omega]$ its corresponding parabolic bundle. Quasi-stability conjecture. The class $[\omega]$ contains a scalar-flat Kähler metric if and only if the corresponding parabolic bundle is quasi-stable. This would characterize precisely which admissible Kähler classes on blown-up ruled surfaces of genus at least $2$ contain scalar-flat Kähler metrics, extending known results for relatively minimal ruled surfaces.