Superexponential chromatic evaluation spectrum at -1

For each integer $n\geq 1$, let $\mathsf{S}_n(-1)=\{P_G(-1):G\text{ is a simple graph on }n\text{ vertices}\}$. Since $|P_G(-1)|$ counts acyclic orientations of $G$, the spectrum can in principle have an upper bound as large as $|P_{K_n}(-1)|=n!$. Superexponential spectrum conjecture. The cardinality $|\mathsf{S}_n(-1)|$ is superexponential in $n$. This asks whether the distinct numbers of acyclic orientations, viewed through chromatic-polynomial evaluations, occur in superexponentially many values; the paper leaves the question open.