The seven-hole conjecture for reduced clique graphs

A chordal graph is a graph with no induced cycle of length at least four. For a graph $G$, let $C_R(G)$ denote its reduced clique graph, whose vertices represent the maximal cliques of $G$ with adjacency defined by nonempty intersection after reduction.

Seven-hole conjecture. There is no chordal graph $G$ such that $C_R(G)$ contains an induced cycle with seven or more vertices.

Reduced clique graphs can have induced cycles of length six, so the conjecture asserts that seven is the largest possible length. It is presented as an open problem; no resolution is given in the source.