The polynomial-time recognition conjecture for reduced clique graphs of chordal graphs

For a graph $G$, let $C_R(G)$ denote its reduced clique graph. A graph is chordal if it has no induced cycle of length at least four. The recognition problem asks whether a given graph is isomorphic to $C_R(G)$ for some chordal graph $G$.

Reduced clique graph recognition conjecture. There is a polynomial-time algorithm for deciding whether a given graph is isomorphic to $C_R(G)$ for some chordal graph $G$.

Polynomial-time recognition is known for clique graphs of chordal graphs, but the source says that the corresponding techniques do not obviously extend to reduced clique graphs. The conjectured algorithm is therefore presented as an open problem.