rank of a matrix with Kronecker product

Question

Let $e_j$ be the $j$th column of $I_n$ with $j=1,...,n$. Also, let $A$ and $B$ be $m \times p$ and $m \times r$ matrices, respectively. Then, what is the rank of $X$ given by \begin{equation} X=[e_i \otimes A, \ B \otimes e_j]. \end{equation} My conjecture is $rank(X)=rank([A, \ B])=rank(A)+rank(B)$ since it seems that $(e_i \otimes A)$ and $(B \otimes e_j)$ will not span the same column space. But I'm not sure and don't know how to prove...