Hendry's cycle-extendibility conjecture for Hamiltonian chordal graphs

Let $G$ be a simple, finite, connected, undirected graph. A graph is Hamiltonian if it contains a cycle containing every vertex, and it is chordal if it contains no induced cycle of length at least $4$. A cycle $C$ is cycle extendible if, whenever $C$ is non-Hamiltonian, there is a cycle $C'$ with $V(C)\subset V(C')$ and $|V(C')|=|V(C)|+1$. A graph is fully cycle extendible if it is cycle extendible and every vertex lies in a triangle.

Hendry's conjecture. Every Hamiltonian chordal graph is fully cycle extendible.

The conjecture was disproved; in particular, the paper studies counterexamples under stronger graph-theoretic conditions, including strong chordality and high connectivity.