Completely independent spanning trees in some Cartesian product graphs

Let $T_1,T_2,\dots ,T_k$ be spanning trees of a graph $G$. For any two vertices $u,v$ of $G$, if the paths from $u$ to $v$ in these $k$ trees are pairwise openly disjoint, then we say that $T_1,T_2,\dots ,T_k$ are completely independent. Hasunuma showed that there are two completely independent spanning trees in any 4-connected maximal planar graph, and that given a graph $G$, the problem of deciding whether there exist two completely independent spanning trees in $G$ is NP-complete. In this paper, we consider the number of completely independent spanning trees in some Cartesian product graphs such as $W_m◻P_n,W_m◻C_n,K_{m,n}◻P_r,K_{m,n}◻C_r,K_{m,n,r}◻P_s,K_{m,n,r}◻C_s$.