Degree-list Kempe equivalence for 3-connected graphs

In a proper coloring, a Kempe change interchanges two colors on a connected component of the subgraph they induce. For a fixed list assignment L, a Kempe change is L-valid if it transforms an L-coloring into another L-coloring. We prove that if G is a $3$-connected graph other than a complete graph or the triangular prism, and L is a list assignment with |L(v)|≥ dG(v) for every vertex v, then G has a proper L-coloring and any two such colorings can be transformed into one another by a finite sequence of L-valid Kempe changes. This proves a conjecture of Chakraborty, Feghali and Mahmoud.