Hamel–Nadirashvili's punctured-disk circular-flow conjecture

Let $D$ be an open non-empty disk of radius $R>0$ centered at the origin, and let $n$ be the outward unit normal on $\partial D$. A flow $\mathbf{v}$ is circular if $\mathbf{v}(x)$ is parallel to $\mathbf{e}_\theta(x)=\left(-x_2/|x|,x_1/|x|\right)$ for every $x\in D\backslash\{0\}$. Let $z\in D$ and let $\mathbf{v}$ be a $C^2(\overline{D}\backslash\{z\})$ and bounded flow solving the stationary two-dimensional Euler equations in $D\backslash\{z\}$, with $\mathbf{v}\cdot n=0$ on $\partial D$.

Hamel–Nadirashvili's conjecture. If $|\mathbf{v}|>0$ in $\overline{D}\backslash\{z\}$, then $z$ is the origin and $\mathbf{v}$ is a circular flow.

This conjecture concerns whether a nonvanishing bounded steady flow in a disk punctured at one point must have its puncture at the center and be circular. The source describes it as difficult to prove and attributes it to Hamel and Nadirashvili.