Abstract In this paper, for p ≥ 4 , we prove that the family of forbidden induced subgraphs for a graph G such that the clique graph of G is Kp-free is finite, providing a proof of a conjecture in [F. Protti and J. Szwarcfiter. Clique-Inverse Graphs of K3-free and K4-free Graphs, Journal of Graph Theory (2000) 257–272]. We also give an upper bound for the rank of Sperner conformal hypergraphs that contain p pairwise intersecting edges and are minimal with respect to all these properties.