In 2008, Deift conjectured that almost periodic initial data leads to almost periodic solutions to the Korteweg-de Vries equation (KdV). In this talk, we show that this is not always the case. Namely, we construct almost periodic initial data whose KdV evolution remains bounded but loses almost periodicity at a later time, by building on the new observation that the conjecture fails for the Airy equation. This is joint work with Rowan Killip and Monica Visan