On the degree of Hilbert polynomials associated to the torsion functor

Let $R$ be a local, Noetherian ring and $I\subseteq R$ an ideal. A question of Kodiyalam asks whether for fixed $i > 0$, the polynomial giving the $i$th Betti number of $I^n$ 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 $\mathfrak{m}$ or $I$ is an integrally closed $\mathfrak{m}$-primary ideal.