The harmonic algebra algebraic-properties conjecture

Let $P$ be a lattice polytope, and let ${\mathcal H}_P\subseteq\mathbb{R}[y_0,\mathbf{y}]$ be its harmonic algebra, with bigraded ideal $\overline{{\mathcal H}}_P$. Harmonic algebra conjecture. The algebra ${\mathcal H}_P$ is

• a Noetherian, equivalently finitely generated, $\mathbb{R}$-subalgebra of $\mathbb{R}[y_0,\mathbf{y}]$;
• Cohen–Macaulay; and
• an algebra whose canonical module $\Omega {\mathcal H}_P$ is isomorphic to the ideal $\overline{{\mathcal H}}_P$, up to a shift in the $\mathbb{N}^2$-grading.

The conjecture is intended to parallel the known properties of the affine semigroup ring associated with $P$ and to explain much of the conjectured structure of the $q$-Ehrhart series. The source supplies no evidence of a resolution, so the conjecture remains open.