Product of matrix valued truncated Toeplitz operators

Let $A_\Phi$ be a matrix valued truncated Toeplitz operator -- the compression of multiplication operator to the vector valued model space $H^2insert ignore into journalissuearticles values(E);\ominus \Theta H^2insert ignore into journalissuearticles values(E);$, where $\Theta$ is a matrix valued non constant inner function. Under supplementary assumptions, we find necessary and sufficient condition that the product $A_\Phi A_\Psi$ is itself a matrix valued truncated Toeplitz operator.