Non-membership conjecture for the sign-imbalance problem in #P

For a poset $P$, its sign imbalance is the absolute difference between the numbers of even and odd linear extensions. The computational problem Sign Imbalance takes a poset $P$ as input and outputs its sign imbalance; the class $\mathsf{\#P}$ consists of functions counting accepting paths of nondeterministic polynomial-time computations. Sign Imbalance non-membership conjecture.

$$\textsc{Sign Imbalance}\notin\mathsf{\#P}.$$

Sign imbalance is naturally in the counting-difference class $\mathsf{GapP}$, but the source states that its non-membership in $\mathsf{\#P}$ remains open.