Parabolic stability necessity conjecture for scalar-flat Kähler iterated blow-ups

Let $E\to\Sigma$ be a rank-$2$ parabolic holomorphic bundle with rational weights over a Riemann surface, and suppose that the ruled surface $\mathbb{P}(E)$ has no non-trivial holomorphic vector field. Let $\widehat M$ be the corresponding iterated blow-up of $\mathbb{P}(E)$.

Parabolic stability necessity conjecture. If $\widehat M$ admits a scalar-flat Kähler metric, then $E$ must be parabolically stable. This is a necessary-condition conjecture for the existence of scalar-flat Kähler metrics on the specified iterated blow-ups; the source gives no proof or resolution.