Mixedness in the Bell violation versus entanglement of formation

Recently, Munro. Nemoto. and White (The Bell Inequality:A Measure of Entanglement?, quant-ph/0102119) tried to indicate that the reason behind a state rho having a higher amount of entanglement (as quantified by the entanglement of formation) than a state rho', but producing the same amount of Bell violation, is due to the fact that the amount of mixedness (as quantified by the linearized entropy) in rho is higher than that in rho'. We counter their argument with examples. We extend these considerations to the von Neumann entropy. Our results suggest that the reason as to why equal amount of Bell violation requires different amounts of entanglement cannot, at least, be explained by mixedness alone.