The conductivity convergence conjecture for the outer problem

Let $D$ be a fixed core, let $V_0$ be fixed, and let $\sigma_c$ denote the core conductivity. Denote by $\Omega$ the solution of the outer problem.

Conductivity convergence conjecture. For fixed $V_0$ and $D$, the solution $\Omega$ of the outer problem converges to a ball as $\sigma_c\to 1$.

Numerical results suggest that the outer problem is well approximated by a one-phase Serrin problem when the core conductivity is close to that of the surrounding medium. The conjecture is supported by the decay of the coefficients in the local derivative calculation, but the paper explains that this is not a rigorous proof without a uniform existence estimate.