Every positive harmonic ball is a subharmonic ball

Let $K$ be a domain, let $x^0$ be a point, and let $\beta$ be a positive measure. A positive harmonic ball is a set of the form $D(x^0,\alpha)$ satisfying the defining harmonic-ball inequality for center $x^0$ and size $\alpha$, while a subharmonic ball is the corresponding set satisfying the subharmonic-ball condition. Positive harmonic ball conjecture. Every positive harmonic ball is a subharmonic ball. The claim concerns the relationship between two classes of quadrature-type domains; the surrounding argument establishes related implications under additional geometric hypotheses, but no resolution of this general statement is given here.