Shnirelman's compact-orbit conjecture for generic 2D Euler limit sets

Let $(\omega,u)$ be a solution of the two-dimensional incompressible Euler equation on $\mathbb{T}^2$, and consider its weak limit set as $t\to\pm\infty$. A solution lies on a compact orbit if its orbit under the Euler evolution is compact in the relevant phase-space topology. Shnirelman's conjecture. The weak limit set of generic solutions consists only of solutions lying on compact orbits. This conjecture is a proposed description of the generic long-time behavior of two-dimensional Euler flows; the supplied source gives no resolution status.