List-colouring the square of a K4K4-minor-free graph

Let G   be a K4K4-minor-free graph with maximum degree ΔΔ. It is known that if Δ∈{2,3}Δ∈{2,3} then G2G2 is (Δ+2)(Δ+2)-degenerate, so that χ(G2)⩽ch(G2)⩽Δ+3χ(G2)⩽ch(G2)⩽Δ+3. It is also known that if Δ⩾4Δ⩾4 then G2G2 is (⌊32Δ⌋+1)-degenerate and χ(G2)⩽⌊32Δ⌋+1. It is proved here that if Δ⩾4Δ⩾4 then G2G2 is ⌈32Δ⌉-degenerate and ch(G2)⩽⌊32Δ⌋+1. These results are sharp.