The small-core convergence conjecture for the outer problem

Let $D$ be the core and let $V_0$ be fixed. Denote by $\Omega$ the solution of the outer problem.

Small-core convergence conjecture. For fixed $V_0$, the solution $\Omega$ of the outer problem converges to a ball as the diameter of $D$ tends to $0$.

This conjecture is motivated by numerical evidence that the outer problem approaches a one-phase Serrin problem when the core is sufficiently small. The paper indicates that the available local analysis does not provide a rigorous proof, because the relevant existence threshold may not be uniform as the core shrinks.