Harmonicity conjecture for the limiting solution outside the support of the datum

Let $cOmega$ be a bounded domain, let $comegasubsetsubsetcOmega$ be the region where the nonnegative datum is supported, and let $u$ be the function given by Theorem 1 with $M(x)\equiv I$. Harmonicity conjecture. The function $u$ is a classical solution of

$$\begin{cases} -\Delta u = 0 & \text{in $\Omega \setminus \overline{\omega}$,} \\ u = 1 & \text{on $\partial\omega$,} \\ u = 0 & \text{on $\partial\Omega$.} \end{cases}$$

This conjecture extends the one-dimensional observation to dimensions $N>1$: outside the region supporting the nonnegative datum, the limiting solution should be harmonic with boundary values $1$ on the inner boundary and $0$ on the outer boundary. The source presents this as an open problem and gives no resolution.