I am looking at a problem in information theory and I have a result that concerns $n$-bit strings of some finite alphabet for $n\in\mathbb{N}$.
I have a distribution $p_{X^n}$ which is some probability distribution over all $n$-bit strings. $p_{X^n}$ is the optimal input distribution to achieve the one-shot capacity of $n$ i.i.d. copies of some channel $W_{Y|X}$ i.e. the one-shot capacity of $W^{\otimes n}_{Y|X}$. $p_{X^n}$ is therefore invariant under any permutation of the $n$ positions. For example, if we consider $n=3$ i.e. $3$-bit strings, I have that $p_{x_1x_2x_3} = p_{x_1x_3x_2} = p_{x_2x_1x_3} = p_{x_2x_3x_1} = p_{x_3x_2x_1} = p_{x_3x_1x_2}$. As we increase $n$, we add more copies of $W_{Y|X}$. Hence, the permutation invariance of our capacity-achieving $p_{X^n}$ holds for any choice of $n\in\mathbb{N}$.
Let $k\in\mathbb{N}$. Let us consider i.i.d. distributions $q_{X^k}^i = \prod\limits_{j=1}^kq^i_{X_j}$ for any choice of distribution $q^i$ on the alphabet. Does there always exist a convex combination of such i.i.d. distributions such that
$$p_{X^k} = \sum_{i}\mu(i)q_{X^k}^i$$
where $\mu(i)$ is some measure that assigns a weight to each element of our convex combination. Is this a claim that follows from the de Finetti theorem (in particular, Theorem 2 of these notes)? The point that I am confused about is whether my $p_{X^k}$ can be thought of as extendible for $n>k$ to allow me to invoke the de Finetti theorem.