EDIT: Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with smooth connected boundary. Let $f\colon \mathbb{R}^3\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^3\backslash \bar\Omega$ and constant on $\partial \Omega$.
Let us assume that $f$ decays at infinity as $O(1/r^2)$.
Is it true that $f\equiv 0$?
This question is a strengthened version of this one Harmonic functions in infinite domain in Euclidean space which contained an unnecessary assumption.
The motivation to ask this question comes from a classical question in electrostatics. Assume that the domain $\Omega$ is filled in with a conductor which is electrically neutral. Can the charge of the conductor redistribute in a non-trivial way so that the conductor will be in equilibrium?