Mixing conjecture for the KdV flow

Let $U(t)$ denote the Koopman operator associated with the KdV flow on the underlying probability space, acting on observables by composition with the flow, and let the projection onto constants be the orthogonal projection onto the constant functions. Mixing conjecture. The KdV flow is mixing: as $t\to\infty$, the operators $U(t)$ converge to the projection onto constants in the weak operator topology. This is the paper's stated long-term ambition for the KdV dynamics; the supplied text does not provide a resolution of the conjecture.