Transpose duality conjecture for preorder multichain matrices

Let $\tau$ be a preorder, and for $k,\ell\in\mathbb N$ let $\nabla_\tau(k,\ell)$ denote the number of $k$-multichains in the poset of lattice points of $\ell\mathcal Q_\tau$. Transpose duality conjecture. The matrix $\nabla_\tau$ is the transpose of $\nabla_{\tau^\ast}$ for every preorder $\tau$. The conjecture generalizes the corresponding identity for preorder polytopes and has been verified by computer on finite pieces for preorders of size at most $5$; its general case remains open.