The recursive term-order conjecture for lecture-hall Gröbner bases

Let $I_n$ be the toric ideal defining the semigroup algebra $\mathbb{C}[L_n\cap\mathbb{Z}^n]$. Order its variables lexicographically according to the recursively arranged Hilbert basis generators: when passing from $n-1$ to $n$, copy each previous block and place $\begin{array}{c}n \\ n-1\end{array}$ above the copied columns.

Recursive term-order conjecture. For this choice of term order, the recursive Gröbner basis conjecture for $I_n$ is true.

The proposed order is intended to realize the recursive structure and Sperner-pair count in the preceding Gröbner basis conjecture. The claim is based on computational evidence for $n\leq 7$ and remains unproved.