The Gorenstein characterization conjecture for s-lecture hall polytopes

Let ${\boldsymbol s}=(s_1,s_2,\ldots,s_n)$ be any sequence, and let $\overleftarrow{\boldsymbol s}$ denote its reverse. Let $\mathbf{P}_n^{({\boldsymbol s})}$ be the corresponding $s$-lecture hall polytope. Gorenstein characterization conjecture. The polytope $\mathbf{P}_n^{({\boldsymbol s})}$ is Gorenstein if and only if there exist ${\boldsymbol c},{\boldsymbol d}\in\mathbb{Z}^n$ satisfying

$$c_js_{j-1}=c_{j-1}s_j+\gcd(s_j,s_{j+1})$$

and

$$d_j\overleftarrow{s_{j-1}}=d_{j-1}\overleftarrow{s_j}+\gcd(\overleftarrow{s_j},\overleftarrow{s_{j+1}})$$

for $j>1$, with $c_1=d_1=1$. The conjecture is motivated by computational evidence and would extend the current Gorenstein characterization beyond cases where some adjacent entries of ${\boldsymbol s}$ are coprime; the source states that this remains unresolved under its method of proof.