Bipartite matrix-valued tensor product correlations that are not finitely representable

We consider the matrix-valued generalizations of bipartite tensor product quantum correlations and bipartite infinite-dimensional tensor product quantum correlations, respectively. These sets are denoted by $C_q^{(n)}(m,k)$ and $C_{qs}^{(n)}(m,k)$, respectively, where $m$ is the number of inputs, $k$ is the number of outputs, and $n$ is the matrix size. We show that, for any $m,k \geq 2$ with $(m,k) \neq (2,2)$, there is an $n \leq 4$ for which we have the separation $C_q^{(n)}(m,k) \neq C_{qs}^{(n)}(m,k)$.