Let $\phi : M_n(\mathbb{C})\to M_m(\mathbb{C})$ be a linear map. $\phi$ is called $k$-positive if the map $\phi^{(k)} : M_{kn}(\mathbb{C}) \to M_{km}(\mathbb{C})$, defined by evaluating $\phi$ entrywise, takes positive matrices to positive matrices.
It is a well-known theorem of Choi that if $\phi$ is $n$-positive, then $\phi$ is actually $k$-positive for all $k$. Is this result optimal? I.e., is there, for each $n$, a map $\phi : M_n(\mathbb{C})\to M_m(\mathbb{C})$ which is $(n-1)$-positive but not $n$-positive? ($m$ can be whatever you want; dependent on $n$, say.)