Suppose that $M$ is a von Neumann algebra with no minimal projections. Let $p$ be a nonzero projection in $M$ and $\rho$ be a normal state on $M$.

For any $\epsilon>0$, can we find a projection $e$ in $M$ such that $0\leq e\leq p$ and $\rho(e)=\epsilon \rho(p)$?