List Colouring Squares of Planar Graphs

In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $\Delta\ge8$ is at most $\lfloor\frac32 \Delta\bigr+1$. We show that it is at most $\frac32 \Delta (1+o(1))$, and indeed this is true for the list chromatic number and for more general classes of graphs.