Parabolicity conjecture of $F$-isocrystals

In this article we prove Crew’s parabolicity conjecture of $F$-isocrystals. For this purpose, we introduce and study the notion of $†$-hull of a sub-$F$-isocrystal. On the way, we prove a new Lefschetz theorem for overconvergent $F$-isocrystals.