de Finetti's theorem roughly states that infinite sequence of exchangeable random variables are conditionally independent. I am looking for tight bounds for de Finetti's theorem in the following scenario.

Suppose the random variable $X_i$ is drawn from $[n] = \{1, \cdots, n\}$ for all $1 \le i \le m$ (not necessarily i.i.d). Further suppose that the sequence $X_1, \cdots, X_m$ is exchangeable meaning that $$ \mathbb{P}((X_1, \cdots, X_m)) = \mathbb{P}((X_{\sigma(1)}, \cdots, X_{\sigma(m)})) $$ for any permutation $\sigma$.

Are there tight bounds known on the distance (in total variation) between the distribution of the sequence $(X_1, \cdots, X_m)$ and the closest mixture of product distributions? In particular, I am interested in bounds that are tight on the size of $|S|$.

I have found only one paper that deals with this issue which is this paper by Diaconis and Freedman. Theorem 3 in this paper gives a distance between the distribution of such a sequence mentioned above and the closest product distribution but it is not mentioned if the dependence on $|S|$ in their result is necessary. I would appreciate any references that deal with my situation.