Cutoff for the Bernoulli-Laplace urn model with $o(n)$ swaps

We study the mixing time of the $(n,k)$ Bernoulli–Laplace urn model, where $k\in \{0,1,\dots ,n\}$. Consider two urns, each containing $n$ balls, so that when combined they have precisely $n$ red balls and $n$ white balls. At each step of the process choose uniformly at random $k$ balls from the left urn and $k$ balls from the right urn and switch them simultaneously. We show that if $k=o(n)$, this Markov chain exhibits mixing time cutoff at $\frac{n}{4k}\log n$ and window of the order $\frac{n}{k}\log \log n$. This is an extension of a classical theorem of Diaconis and Shahshahani who treated the case $k=1$.