Duality conjecture for transmuted preorder M-triangles

Let $\tau$ be a preorder of size $n$, let $P_\tau$ be its lattice-point poset, and let $\overline M_{P_\tau}$ be the transmuted $M$-triangle. Transmuted M-triangle duality conjecture.

$$(xy)^n\overline M_{P_\tau}(1/x,1/y)=\overline M_{P_\tau}(y,x).$$

Equivalently, if $L_\tau$ is a lattice realization as in the preceding conjecture, then $M_{L_\tau}(x,y)=M_{L_\tau^\ast}(x,y)$, where $L_\tau^\ast$ is the dual lattice. The source provides no resolution.