Generic sequences of polynomials

Let I be an ideal in a polynomial ring S over an infinite field such that I is generated by a generic sequence of homogeneous polynomials of specified degrees. Fröberg has conjectured a formula for the Hilbert series of S/I. Moreno-Socías has conjectured a combinatorial property for the initial ideal of I with respect to degree reverse lexicographic order. I show that Moreno-Socías' Conjecture implies Fröberg's Conjecture. I also give a criterion for a Hilbert series to admit an ideal with the property proposed by Moreno-Socías and show that the Hilbert series proposed by Fröberg does have this property.