Let $M$ and $N$ be von Neumann algebras equipped with faithful normal states. Let $\pi \colon M \to N$ be a normal unital injective $*$-homomorphism preserving the states and commuting with the modular groups. I know that in this situation there exists a faithful normal conditional expectation from $N$ onto $\pi(M)$ preserving the states.
Suppose that $N$ is injective. Is $M$ injective ?