Independence of multispins in rooted forests of Cartesian products of complete graphs

Let $G=(U,A)$ be a weighted digraph, let $K_p$ be the complete graph with vertex set $[p]$, and let $H=G\times K_p$ be their Cartesian product. A rooted forest of $H$ has a $G$-projection if it records the number of $a$-arcs for every $a\in A$ and the number of vertical arcs at every $u\in U$. For a rooted forest $F$ and $u\in U$, let $\sigma_u$ denote the multiset of spins of the vertical arcs of $F$ at $u$. Given a rooted forest $F_0$ of $H$, choose $F$ uniformly among rooted forests of $H$ having the same $G$-projection as $F_0$. Independence of multispins. The multispins $(\sigma_u)_{u\in U}$ of $F$ at the different vertices of $G$ are independent.