Given a chordal graph $G$, what is the complexity of computing the reduced clique graph $C_r(G)$?

A graph $G$ is chordal if it doesn't have induced cycles of length $4$ or more. A clique tree $T$ of $G$ is a tree in which the vertices of the tree are the maximal cliques of $G$. An edge in $T$ corresponds to a minimal separator. The number of distinct clique trees can be exponential in the number of vertices in a chordal graph.

The reduced clique graph $C_r(G)$ is the union of all clique trees of $G$. That is, it has all the same vertices, and all possible edges. What is the complexity of computing $C_r(G)$ for a given $G$?

I think I once saw a presentation claiming $C_r(G)$ can be computed in $O(m+n)$ time without proof. This would mean it is as easy as computing a clique tree of $G$. Is there a reference that confirms this, or gives a slower algorithm for computing it?