Let $I$ be an ideal in a polynomial ring $S = k[x_1, \ldots, x_n]$, where $k$ is an algebraically closed field of characteristic zero. Fix a term order on $S$ (e.g. a lexicographic order) and let $in(I)$ denote the initial ideal of $I$, that is, the monomial ideal generated by all the initial terms of elements of $I$.
It is clear that for any $k > 0$ one has $in(I)^k \subset in(I^k)$, with equality if $I$ itself is a monomial ideal.
Is there an example of $I$ where for large $k$, the quotient $in(I^k) / in(I)^k$ is NOT finite dimensional (over $k$)?
Is there a "good" sufficient condition on $I$ and the term order to guarantee that for large $k$, $in(I^k) / in(I)^k$ is always finite dimensional? Clearly if $I$ is primary or is a monomial ideal this is the case.