Semistable reduction for overconvergent F-isocrystals on a curve

Given a smooth affine curve X over a field k of positive characteristic, and an overconvergent F-isocrystal on X, we prove after replacing k by a finite purely inseparable extension, there exists a finite separable cover of X, the pullback of the isocrystal along which extends to a log-F-isocrystal on a smooth compactification. This resolves a weak form of the global version of a conjecture of Crew; the proof uses the local version of the conjecture, established separately by Andre, Mebkhout and the author.