  Let $\mathscr{A}$ be a unital $C^*$-algebra and let $\Phi: \mathscr{A} \to
{\mathbb B}({\mathscr H})$ be a unital $n$-positive linear map between
$C^*$-algebras for some $n \geq 3$. We show that $$\|\Phi(AB)-\Phi(A)\Phi(B)\|
\leq \Delta(A,||\cdot||)\,\Delta(B,||\cdot||)$$ for all operators $A, B \in
\mathscr{A}$, where $\Delta(C,\|\cdot\|)$ denotes the operator norm distance of
$C$ from the scalar operators.
