Let k be a finite field of characteristic p, and let l be a prime not equal to p. An old conjecture of Deligne states that on any smooth variety over k, any lisse l-adic sheaf which is irreducible of finite determinant admits a "crystalline companion" with matching Frobenius characteristic polynomials at all closed points. We report on progress towards constructing crystalline companions in the category of overconvergent F-isocrystals, building on work of Drinfeld, Deligne, and Abe-Esnault.