Landis-type uniqueness-at-infinity conjecture for periodic elliptic equations

Let $u$ be a super-exponentially decaying solution of a second-order periodic, or more generally an appropriate class of periodic elliptic, equation. Landis-type conjecture. Such a solution should be prohibited, at least in the self-adjoint case, by a form of uniqueness of continuation at infinity. Consequently, the corresponding periodic elliptic operators should have absolutely continuous spectrum. This is proposed as a route to proving absolute continuity: a bound state would yield a super-exponentially decaying solution, while a suitable Landis-type uniqueness theorem would rule it out.