Clique-root conjecture for l-connected chordal K(l+3)-free graphs

Let $l$ be a positive integer, and let $G$ be an $l$-connected chordal graph. Let $K_{l+3}$ denote the complete graph on $l+3$ vertices. A clique root is a real root of the clique polynomial $C(G,x)$.

Clique-root conjecture. If $G$ is $K_{l+3}$-free, then $G$ has only clique roots.

This is presented as a stronger conjecture extending the preceding claims for restricted connectivity and clique-exclusion classes. The source does not state whether it has been resolved.