If any order preserving permutation submatrix of $A\otimes B$ belongs to $\{A,B\}$ does $A=B$?

Question

Edited after the comment below: let $A$ and $B$ be two permutation matrices of size $n$. If any order preserving combination of $n$ rows and $n$ columns of $A\otimes B$ that is a permutation matrix is equal to $A$ or to $B$, does $A=B$? If yes, is it either the identity matrix or the one consisting only of $1$ on the secondary diagonal? Here $\otimes$ is the Kronecker product.