  Let $X$ be a smooth scheme over a finite field. It is conjectured that a
convergent $F$-isocrystal on $X$ is overconvergent if its restriction to every
curve contained in $X$ is overconvergent. Using the theory of \'etale and
crystalline companions, we establish a weaker version of this criterion in
which we also assume that the wild local monodromy of the restrictions to
curves is trivialized by pullback along a single dominant morphism to $X$.
