Persistence of Cat's eyes for all rotation parameters

Let $D$ be a rotating vortex patch, let $\Psi$ be its stream function, and let $s>0$ be the parameter in the family of patches. For sufficiently small $s$, Theorem~ establishes a phase portrait for $\dot z=\nabla^\perp\Psi(z)$ outside $D$ consisting of $m$ saddle points, heteroclinic connections enclosing periodic-orbit regions around $m$ centers, with all other orbits given by polar graphs.

Persistence-of-Cat's-eyes conjecture. The conclusion of Theorem~ holds for all $s>0$.

The conjecture extends the rigorously established small-$s$ Cat's-eyes structure to the entire global bifurcation curve. The source gives no evidence of a resolution beyond the small-parameter theorem and numerical motivation.