Hamel–Nadirashvili's one-stagnation-point 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 $\mathbf{v}\in C^2(\overline{D})$ solve the stationary two-dimensional Euler equations in $D$, with $\mathbf{v}\cdot n=0$ on $\partial D$.

Hamel–Nadirashvili's weaker conjecture. If $z\in D$ is the only stagnation point of $\mathbf{v}$ in $\overline{D}$, meaning that $|\mathbf{v}(z)|=0$ and $|\mathbf{v}|>0$ in $\overline{D}\backslash\{z\}$, then $z$ is the origin and $\mathbf{v}$ is a circular flow.

This weaker version is presented as more promising than the punctured-disk conjecture, while still asserting that a steady flow with exactly one interior stagnation point must be centered and circular.