  Generalizing Courant's nodal domain theorem, the "Extended Courant property"
is the statement that a linear combination of the first $n$ eigenfunctions has
at most $n$ nodal domains. In a previous paper (Documenta Mathematica, 2018,
Vol. 23, pp. 1561--1585), we gave simple counterexamples to this property,
including convex domains. In the present paper, using some input from numerical
computations, we pursue the investigation of the Extended Courant property with
two new examples, the equilateral rhombus and the regular hexagon.
