  In this paper, we prove that if the dual of a Bruhat interval in a Weyl group
is a zircon, then that interval is rationally smooth. Investigating when the
converse holds, and drawing inspiration from conjectures by Delanoy, leads us
to pose two conjectures. If true, they imply that for Bruhat intervals in type
$A$, duals of smooth intervals, zircons, and being isomorphic to lower
intervals are all equivalent. As a verification, we have checked our
conjectures in types $A_n$, $n\leq 8$.
