Kohl–Olsen levelness conjecture for lecture hall simplices

Let ${\boldsymbol s}\in\mathbb{Z}_{\geq 1}^n$ be a sequence. Suppose there exists ${\boldsymbol c}\in\mathbb{Z}^n$ such that

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

for every $j>1$, with $c_1=1$. Kohl–Olsen's levelness conjecture. The ${\boldsymbol s}$-lecture hall simplex $\mathbf{P}_n^{({\boldsymbol s})}$ is level. The conjecture supplies a sufficient condition for levelness; the source gives no resolution and discusses the difficulty of finding practical characterizations.