q-zeta reciprocity conjecture for preorder lattice-point posets

Let $\tau$ be a preorder of size $n$, let $P_\tau$ be the poset of lattice points associated with its preorder polytope, and let $\mathcal Z_q(P_\tau,t)$ be its $q$-zeta polynomial. Write $\operatorname{Max}(P_\tau)$ for the set of maximal elements, let $\operatorname{cov}(\mathbf a)$ be the number of elements covered by $\mathbf a$, and set $[-1]_q=-1/q$. q-zeta reciprocity conjecture.

$$\mathcal Z_q(P_\tau,[-1]_q)=(-1)^n\sum_{\mathbf a\in\operatorname{Max}(P_\tau)}q^{-\operatorname{cov}(\mathbf a)}.$$

In particular,

$$\mathcal Z(P_\tau,-1)=(-1)^n\#\operatorname{Max}(P_\tau).$$

The statement is motivated by computational evidence; its general validity is open.