On the degree of Hilbert polynomials associated to the torsion functor

Let R be a local, Noetherian ring and I. R an ideal. A question of Kodiyalam asks whether for fixed i > 0, the polynomial giving the ith Betti number of In has degree equal to the analytic spread of I minus one. Under mild conditions on R, we show that the answer is positive in a number of cases, including when I is divisible by m or I is an integrally closed m-primary ideal.