We report on joint work-in-progress with Ambrus Pal on the following conjecture. Conjecture: Let X be a smooth variety over a finite field k and let E be an overconvergent F-isocrystal on X that has rank 2 and is absolutely irreducible. Suppose further that E has "infinite monodromy at a divisor at infinity." Then E comes from a family of abelian varieties.