Wilfong–Haxell–Winkler delay-colouring conjecture for bipartite multigraphs

Let $G$ be a bipartite multigraph with partition classes $A,B$ and maximum degree $d$. Each edge $e$ has an associated integer delay $r(e)$. An edge-colouring is a map $f:E(G)\to\{0,\ldots,d\}$, and $f+r\pmod{d+1}$ is defined on an edge $e$ as $f(e)+r(e)\pmod{d+1}$. Wilfong–Haxell–Winkler's delay-colouring conjecture. The graph $G$ admits an edge-colouring $f$ such that $f$ is proper on $A$ and $f+r\pmod{d+1}$ is proper on $B$. This conjecture is motivated by scheduling in optical networks and is proved in the paper in the special case $d=3$; the general case remains open.