A crystalline incarnation of Berthelot's conjecture and K\"unneth formula for isocrystals

Berthelot’s conjecture predicts that under a proper and smooth morphism of schemes in characteristic $p$, the higher direct images of an overconvergent $F$-isocrystal are overconvergent $F$-isocrystals. In this paper we prove that this is true for crystals up to isogeny. As an application we prove a K"unneth formula for the crystalline fundamental group.