Polynomial-time conjecture for counting constrained linear extensions

Let $P$ be a finite poset with fixed prescribed values represented by $\mathbf z,\mathbf c$, and let $N_{\mathbf z,\mathbf c}(P)$ denote the number of linear extensions satisfying those prescriptions. Constrained linear-extension counting conjecture. For every fixed integer $m\in\mathbb N$, the decision problem

$$\{N_{\mathbf z,\mathbf c}(P)=m\}$$

is in ${\mathsf P}$. The source proves the case $m=1$ and conjectures the statement for every fixed $m$.