The levelness conjecture for a family of s-lecture hall polytopes

Let ${\boldsymbol s}\in\mathbb{Z}_{\geq 1}^n$ be a sequence, and let $\mathbf{P}_n^{({\boldsymbol s})}$ denote the corresponding $s$-lecture hall polytope. Levelness conjecture. Suppose there exists ${\boldsymbol c}\in\mathbb{Z}^n$ satisfying

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

for $j>1$, with $c_1=1$. Then $\mathbf{P}_n^{({\boldsymbol s})}$ is level. This is proposed on the basis of experimental evidence as a potentially tractable sufficient condition for levelness in a large family of lecture hall polytopes; no resolution is supplied in the source.