McDiarmid–Reed's conjecture for hexagonal graphs

A hexagonal graph is a finite triangle-free induced subgraph of the triangular lattice. A graph is $(a,b)$-colorable if each vertex can be assigned a set of $b$ colors from a common set of $a$ colors so that adjacent vertices receive disjoint sets.

McDiarmid–Reed's conjecture. Every hexagonal graph is $(9,4)$-colorable.

This conjecture was proposed by McDiarmid and Reed in 1999 and is the central coloring problem addressed by the paper. The paper develops new reducible configurations and reports experimental progress, but does not establish the conjecture.