  A long-standing open conjecture in combinatorics asserts that a Gorenstein
lattice polytope with the integer decomposition property (IDP) has a unimodal
(Ehrhart) $h^\ast$-polynomial. This conjecture can be viewed as a strengthening
of a previously disproved conjecture which stated that any Gorenstein lattice
polytope has a unimodal $h^\ast$-polynomial. The first counterexamples to
unimodality for Gorenstein lattice polytopes were given in even dimensions
greater than five by Musta{\c{t}}{\v{a}} and Payne, and this was extended to
all dimensions greater than five by Payne. While there exist numerous examples
in support of the conjecture that IDP reflexives are $h^\ast$-unimodal, its
validity has not yet been considered for families of reflexive lattice
simplices that closely generalize Payne's counterexamples. The main purpose of
this work is to prove that the former conjecture does indeed hold for a natural
generalization of Payne's examples. The second purpose of this work is to
extend this investigation to a broader class of lattice simplices, for which we
present new results and open problems.
