Weakly localized solutions for localized time-dependent potentials

Consider a time-dependent potential that is localized in space. A weakly localized solution (WLS) is a solution with a component exhibiting weak localization, while a localized solution is localized in space in the stronger sense used in the paper. Weak-localization conjecture. Time-dependent potentials localized in space may have weakly localized solutions which are not localized. The claim concerns the possible existence of nonlocalized weakly localized behavior in time-dependent Schrödinger dynamics; the source gives no resolution and places it among open questions about weakly localized solutions.