The dual-partition refinement conjecture for poset-weight partitions

Let $\boldsymbol{P}=(\Omega,\preccurlyeq_{\boldsymbol{P}})$ be a poset, let $(H_i\mid i\in\Omega)$ be finite abelian groups, and let $\boldsymbol{H}=\prod_{i\in\Omega}H_i$. Let $\widehat{\boldsymbol{H}}$ be the character group of $\boldsymbol{H}$, identified with $\prod_{i\in\Omega}\widehat{H_i}$ by

$$\alpha(\beta)=\prod_{i\in\Omega}\alpha_{(i)}(\beta_{(i)})$$

for $\alpha\in\widehat{\boldsymbol{H}}$ and $\beta\in\boldsymbol{H}$. Define the poset-weight partition $\mathcal{Q}(\boldsymbol{H},\boldsymbol{P})$ using $\operatorname{wt}_{\boldsymbol{P}}(\beta)=|\langle\operatorname{supp}(\beta)\rangle_{\boldsymbol{P}}|$, and let $\overline{\boldsymbol{P}}$ denote the opposite poset. The dual partition of a partition is the partition induced by equality of its character sums.

Dual-partition refinement conjecture. If $|H_i|\geqslant2$ for all $i\in\Omega$, then the dual partition of $\mathcal{Q}(\boldsymbol{H},\boldsymbol{P})$ is finer than $\mathcal{Q}(\widehat{\boldsymbol{H}},\overline{\boldsymbol{P}})$.

This conjecture concerns MacWilliams-type identities for poset weights on finite abelian groups. The parser marks it as resolved, and the supplied evidence states that the relevant result has been established in the cited literature.