  In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle
extendable. After a series of papers confirming the conjecture for a number of
graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015.
Given that their counterexamples are not strongly chordal graphs and they are
all only $2$-connected, Lafond and Seamone asked the following two questions:
(1) Are Hamiltonian strongly chordal graphs cycle extendable? (2) Is there an
integer $k$ such that all $k$-connected Hamiltonian chordal graphs are cycle
extendable?
  Later, a conjecture stronger than Hendry's is proposed. In this paper, we
resolve all these questions in the negative. On the positive side, we add to
the list of cycle extendable graphs two more graph classes, namely, Hamiltonian
$4$-\textsc{fan}-free chordal graphs where every induced $K_5 - e$ has true
twins, and Hamiltonian $\{4\textsc{-fan}, \overline{A} \}$-free chordal graphs.
