Circularity conjecture for bounded Euler flows in a disk

Let $D$ be an open non-empty disk and let $z\in D$. Let $v$ be a $C^2(\overline{D}\!\setminus\!\{z\})$ and bounded flow solving the Euler equations and satisfying $v\cdot n=0$ on $\partial D$, where $n$ denotes the outward unit normal on $\partial D$. Assume that $|v|>0$ in $\overline{D}\!\setminus\!\{z\}$. Circularity conjecture. Then $z$ is the center of the disk and the flow is circular with respect to $z$. This conjecture concerns rigidity of stationary Euler flows in annular domains: under the stated regularity, boundary, boundedness, and non-vanishing assumptions, it predicts that the possible flow must be circular and centered at the puncture. The supplied text gives no evidence of resolution.