  We establish factoriality and non-injectivity in full generality for the
mixed $q$-Araki-Woods von Neumann algebra associated to a separable real
Hilbert space $\mathsf{H}_{\mathbf{R}}$ with $\dim\mathsf{H}_{\mathbf{R}}\geq
2$, a strongly continuous one parameter group of orthogonal transformations on
$\mathsf{H}_\mathbb{R}$, a direct sum decomposition
$\mathsf{H}_{\mathbf{R}}=\oplus_{i}\mathsf{H}_{\mathbb{R}}^{(i)}$, and a real
symmetric matrix $(q_{ij})$ with $q=\sup_{i,j}|q_{ij}|<1$. This is achieved by
first proving the existence of conjugate variables for a finite number of
generators of the algebras, following the lines of Miyagawa-Speicher and
Kumar-Skalski-Wasilewski. The conjugate variables belong to the factors in
question and satisfy certain Lipschitz condition.
