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We prove some unique prime factorization results for tensor products of type$II_1$ factors of the form $\Gamma_q(\mathbb{C}, S \otimes H)$ arising fromsymmetric independent copies with sub-exponential dimensions of the spaces$D_k(S)$ and dim$(H)$ finite and greater than a constant depending on $q$.

Generalized $q$-Gaussian von Neumann algebras with coefficients, III. Unique prime factorization results