Lattice-realization conjecture for preorder lattice-point posets

Let $\tau$ be a preorder of size $n$, let $P_\tau$ be its lattice-point poset, let $M_{P_\tau}(x,y)$ be its $M$-triangle, and let $\overline{M}_{P_\tau}$ be the transmutation defined by

$$\overline f(x,y)=f\left(\frac{1-y}{1-xy},1-xy\right).$$

Lattice-realization conjecture. There exists a finite graded lattice $L_\tau$ of rank $n$ such that

$$\mathcal Z(P_\tau,t)=\mathcal Z(L_\tau,t),\qquad \overline M_{P_\tau}(x,y)=M_{L_\tau}(x,y).$$

In particular, the rank-generating polynomial of $L_\tau$ equals $h(\tau,t)$. The conjecture extends the arbor setting; the source gives examples for chain preorders but no general resolution.