  Let $\phi:M_n\to B(H)$ be an injective, completely positive contraction with
$\V\phi^{-1}:\phi(M_n)\to M_n\V_{cb}\leq1+\delta(\epsilon).$ We show that if
either (i) $\phi(M_n)$ is faithful modulo the compact operators or (ii)
$\phi(M_n)$ approximately contains a rank 1 projection, then there is a
complete order embedding $\psi:M_n\to B(H)$ with $\V\phi-\psi\V_{cb}<\epsilon.$
We also give examples showing that such a perturbation does not exist in
general. As an application, we show that every $C^*$-algebra $A$ with
$\mathcal{OL}_\infty(A)=1$ and a finite separating family of primitive ideals
is a strong NF algebra, providing a partial answer to a question of Junge,
Ozawa and Ruan.
