Hamel–Nadirashvili's rigidity conjecture for steady Euler flows in a punctured disk

Let $D$ be an open disk centered at the origin. Let $z\in D$ and let $\mathbf{v}\in C^2(\overline{D}\backslash \\{z\\})$ be a bounded flow solving the Euler equations on $D\backslash\\{z\\}$ and satisfying $\mathbf{v}\cdot \mathbf{n}=0$ on $\partial D$, where $\mathbf{n}$ denotes the outward unit normal on $\partial D$. Assume that $|\mathbf{v}|>0$ in $\overline{D}\backslash\\{z\\}$. Hamel–Nadirashvili's rigidity conjecture. Then $z$ is the origin and $\mathbf{v}$ is a circular flow. This conjecture concerns whether steady Euler flows inherit the geometric symmetry of a disk. Related rigidity results are known for strips, half-planes, the whole plane, annuli, and several punctured or exterior domains, while this punctured-disk case remains open in the supplied source.