The conjecture that subharmonic quadrature domains satisfy the equivalent conditions of Theorem 1

Let $\Omega$ be a subharmonic quadrature domain as above, with its associated function $u$ and boundary decomposition as in the preceding setup. Subharmonic quadrature-domain conjecture. The domain $\Omega$ satisfies the equivalent conditions in Theorem 1. This conjecture concerns the geometric and analytic characterization of subharmonic quadrature domains developed in the paper; the supplied text does not state which of the equivalent conditions are known or whether the conjecture has been resolved.