Exterior-domain rigidity conjecture for the discrete-time overdetermined heat problem

Let $N\ge 2$ and let $\Omega\subset\mathbb{R}^N$ be an exterior domain, meaning the complement of a bounded closed set. Consider the initial-boundary value problem

$$\begin{cases} \partial_t u=\Delta u & \text{in }\Omega\times(0,\infty),\\ u=1 & \text{on }\Omega\times\{0\},\\ u=0 & \text{on }\partial\Omega\times(0,\infty),\\ u(x,t)\to 0 & \text{uniformly in $t$ as $|x|\to\infty$}. \end{cases}$$

Exterior-domain rigidity conjecture. Complements of closed balls are the only exterior domains whose solution satisfies the discrete-time overdetermined boundary condition.

This is the unbounded-domain analogue of the rigidity problem for the overdetermined heat equation. The conjecture is motivated by continuous-time results obtained using moving planes, but the discrete-time condition and the lack of compactness in unbounded domains make the corresponding classification problem unresolved.