The Fourier-reflexivity characterization of hierarchical posets

Let $\boldsymbol{P}=(\Omega,\preccurlyeq_{\boldsymbol{P}})$ be a poset, let $(H_i\mid i\in\Omega)$ be finite abelian groups, and let $\boldsymbol{H}=\prod_{i\in\Omega}H_i$. For a codeword $\beta\in\boldsymbol{H}$, define

$$\operatorname{supp}(\beta)=\\{i\mid i\in\Omega,~\beta_{(i)}\neq1_{H_i}\\}$$

and

$$\operatorname{wt}_{\boldsymbol{P}}(\beta)=|\langle\operatorname{supp}(\beta)\rangle_{\boldsymbol{P}}|.$$

Let $\mathcal{Q}(\boldsymbol{H},\boldsymbol{P})$ be the partition of $\boldsymbol{H}$ whose blocks are the sets of codewords having equal $\boldsymbol{P}$-weight. The poset $\boldsymbol{P}$ is hierarchical if $u\preccurlyeq_{\boldsymbol{P}}v$ whenever $u,v\in\Omega$ satisfy $\operatorname{len}(u)+1\leqslant\operatorname{len}(v)$.

Fourier-reflexivity characterization. If $|H_i|\geqslant2$ for all $i\in\Omega$ and $\mathcal{Q}(\boldsymbol{H},\boldsymbol{P})$ is Fourier-reflexive, then $\boldsymbol{P}$ is hierarchical.

This conjecture characterizes hierarchical posets through Fourier-reflexivity of their poset-weight partitions. The parser marks it as resolved; the supplied evidence indicates that the result was established in the source's cited literature, so it is recorded as solved.