The explicit bijection conjecture for edges of rook equivalence graphs

Let $B$ be a board, let $G(B)$ be its rook equivalence graph, and suppose that $\{B_1,B_2\}$ is an edge of $G(B)$. For a nonnegative integer $k$, consider the placements of $k$ rooks on $B_1$ and on $B_2$. Explicit rook-placement bijection conjecture. There is an explicit bijection between placements of $k$ rooks on $B_1$ and placements of $k$ rooks on $B_2$. The paper presents this as a future project motivated by the goal of constructing geometrically explicit bijections for rook-equivalent boards; no resolution is supplied here.