The Lecture Hall cone initial-subalgebra conjecture

Let $n\in\mathbb{N}$. There exist a $\mathbb{Z}^2$-graded polynomial ring $\Bbbk[x_1,\dots,x_m]$ for some $m\in\mathbb{N}$, a graded subalgebra $A_n\subseteq\Bbbk[x_1,\dots,x_m]$, and a term order $\prec$ on this polynomial ring such that $A_n$ is isomorphic to a polynomial ring in $n$ variables of degrees $(1,0),(2,1),\dots,(n,n-1)$, and $\operatorname{In}_\prec(A_n)$ is isomorphic to the Ehrhart ring of the Lecture Hall cone $\mathbf{L}_n$. The Lecture Hall cone initial-subalgebra conjecture. For every $n\in\mathbb{N}$, such a graded polynomial-ring realization and term order exist. The Hilbert series of the proposed polynomial ring is $\prod_{i=1}^n(1-q_1^iq_2^{i-1})^{-1}$, matching the Lecture Hall Theorem; the existence of the realization remains conjectural.