A parallel repetition theorem for entangled two-player one-round games under product distributions

We show a parallel repetition theorem for the entangled value $\omega^*(G)$ of any two-player one-round game $G$ where the questions $(x,y) \in \mathcal{X}\times\mathcal{Y}$ to Alice and Bob are drawn from a product distribution on $\mathcal{X}\times\mathcal{Y}$. We show that for the $k$-fold product $G^k$ of the game $G$ (which represents the game $G$ played in parallel $k$ times independently), $\omega^*(G^k) =\left(1-(1-\omega^*(G))^3\right)^{\Omega\left(\frac{k}{\log(|\mathcal{A}| \cdot |\mathcal{B}|)}\right)}$, where $\mathcal{A}$ and $\mathcal{B}$ represent the sets from which the answers of Alice and Bob are drawn.