Conjecture on the detailed shape of the Sadovskii vortex patch

Let $\Omega$ be the bounded open set defining a Sadovskii vortex patch as described in Theorem 1. Detailed-shape conjecture. The set $\Omega$ is connected, convex, and makes a right angle with the horizontal axis. The connectedness is not known because the discontinuous vorticity used in the variational construction is not regular enough for standard connectedness arguments. The right-angle assertion is also unresolved without additional assumptions on boundary regularity; the boundary may oscillate near the touching point.