Counterexamples for Cohen-Macaulayness of Lattice Ideals

Let $\mathscr{L}\subset \mathbb{Z}^n$ be a lattice, $I$ its corresponding lattice ideal, and $J$ the toric ideal arising from the saturation of $\mathscr{L}$. We produce infinitely many examples, in every codimension, of pairs $I,J$ where one of these ideals is Cohen--Macaulay but the other is not.