Deift's almost-periodic dynamics conjecture for the KdV equation

Let $u(t,x)$ solve the KdV equation

$$\partial_tu+\partial_x^3u+u\partial_xu=0.$$

Assume that the initial data $u(0,x)$ is almost periodic in $x$. Deift's conjecture. The solution $u(t,x)$ is almost periodic in time $t$ and retains the same spatial almost periodicity as the initial data for all times. This conjecture concerns the long-time dynamics of KdV solutions with nondecaying almost-periodic initial data; the source presents it as a motivation for studying analogous existence, uniqueness, and asymptotic questions for nonlinear Schrödinger equations. Its resolution is not established in the supplied text.