The complete isomorphism class of an operator space

Suppose $X$ is an infinite-dimensional operator space and $n$ is a positive integer. We prove that for every $C > 0$ there exists an operator space $\tilde{X}$ such that the formal identity map $id : X \to \tilde{X}$ is a complete isomorphism, $I_{\mathbf{M}_n} \otimes id$ is an isometry, and $d_{cb}(X, \tilde{X}) > C$. This provides a non-commutative counterpart to a recent result of W. Johnson and E. Odell.