The complete bipartite rook equivalence graph conjecture for Ferrers boards

Let $K_{a,b}$ be a complete bipartite graph, and let $B$ be a Ferrers board whose rook equivalence graph is $K_{a,b}$. Complete bipartite rook equivalence conjecture. Then $a=b=1$. This conjecture concerns which graphs can arise as rook equivalence graphs of Ferrers boards; the paper presents it as a future project, and no resolution is supplied here.