  A recent conjecture that appeared in three papers by Bigdeli--Faridi,
Dochtermann, and Nikseresht, is that every simplicial complex whose clique
complex has shellable Alexander dual, is ridge-chordal. This strengthens the
long-standing Simon's conjecture that the $k$-skeleton of the simplex is
extendably shellable, for any $k$. We show that the stronger conjecture has a
negative answer, by exhibiting an infinite family of counterexamples.
