Linear maps preserving (p,k) norms of tensor products of matrices

Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices. Let $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1\le k\le mn$ and $2<p<\mathrm{\infty }$. We show that a linear map $\phi :M_{mn}\to M_{mn}$ satisfies $$\mathrm{\parallel }\phi (A\otimes B){\mathrm{\parallel }}_{(p,k)}=\mathrm{\parallel }A\otimes B{\mathrm{\parallel }}_{(p,k)}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}A\in M_m\mathrm{a}\mathrm{n}\mathrm{d}B\in M_n$$ if and only if there exist unitary matrices $U,V\in M_{mn}$ such that $$\phi (A\otimes B)=U({\phi }_1(A)\otimes {\phi }_2(B))V\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}A\in M_m\mathrm{a}\mathrm{n}\mathrm{d}B\in M_n,$$ where ${\phi }_s$ is the identity map or the transposition map $X\to X^T$ for $s=1,2$. The result is also extended to multipartite systems.