Vizing's edge-recoloring conjecture holds

In 1964 Vizing proved that starting from any k-edge-coloring of a graph G one can reach, using only Kempe swaps, a ($\mathrm{\Delta }$ + 1)-edge-coloring of G where $\mathrm{\Delta }$ is the maximum degree of G. One year later he conjectured that one can also reach a $\mathrm{\Delta }$-edge-coloring of G if there exists one. Bonamy et. al proved that the conjecture is true for the case of triangle-free graphs. In this paper we prove the conjecture for all graphs.