Coloring squares of planar graphs with girth six

Wang and Lih conjectured that for every g≥5g≥5, there exists a number M(g)M(g) such that the square of a planar graph GG of girth at least gg and maximum degree Δ≥M(g)Δ≥M(g) is (Δ+1)(Δ+1)-colorable. The conjecture is known to be true for g≥7g≥7 but false for g∈{5,6}g∈{5,6}. We show that the conjecture for g=6g=6 is off by just one, i.e., the square of a planar graph GG of girth at least six and sufficiently large maximum degree is (Δ+2)(Δ+2)-colorable.