The conjecture that every quintic graph has at least two Kempe equivalence classes

Let a quintic graph be a 5-regular graph, and let two 1-factorisations be Kempe equivalent if one can be obtained from the other by a sequence of Kempe switches. Kempe-class conjecture. Every quintic graph contains at least two Kempe equivalence classes. The paper proves the existence of infinitely many planar 5-connected quintic graphs with at least two Kempe equivalence classes, but the asserted statement for all quintic graphs remains open.