The chordal graph representation conjecture for clique and reduced clique graphs

Let $C(G)$ denote the clique graph of a graph $G$, and let $C_R(G)$ denote its reduced clique graph. A graph is chordal if it has no induced cycle of length at least four.

Chordal graph representation conjecture. Let $H$ be a chordal graph. There are chordal graphs $G$ and $G'$ such that $H$ is isomorphic to both $C(G)$ and $C_R(G')$.

The conjecture would establish that every chordal graph is simultaneously a clique graph and a reduced clique graph of chordal graphs. The source reports this as an open conjecture, while noting that a polynomial-time recognition algorithm is known for clique graphs of chordal graphs.