Algebraic Frobenius Structures

Given a family of motives, the de Rham realization (a certain vector bundle with integrable "Gauss-Manin" connection) can be compared to the crystalline realizations for various primes p, but the resulting Frobenius structures cannot be directly compared to each other. We formulate, and give some evidence for, a conjecture asserting that these Frobenius structures "commute" in a suitable sense.