Conjectured decomposition of level-$k$ Dyck vectors

Fix an integer $k geq 0$. For each $n geq 0$ and each $\mu iu a\text{Par}(k)$, let $\mathcal{DV}_{n,k}$ denote the set of level-$k$ Dyck vectors of length $n$. For a partition $\lambda$, $\textsc{dyck}(\lambda)$ is the associated Dyck vector, and $\lambda'$ is its conjugate partition. Two subsets are opposite when corresponding objects have their area and dinv interchanged. The map $\iota_n$ is the injection from the stated inclusion of Dyck-vector sets.

Conjectured decomposition. There exist possibly identical subsets $\mathcal{DV}_{n,\mu}$ and $\overline{\mathcal{DV}}_{n,\mu}$ of $\mathcal{DV}_{n,k}$ such that: (a) each of the two families, indexed by $\mu iu \operatorname{Par}(k)$, is a disjoint partition of $\mathcal{DV}_{n,k}$; (b) $\mathcal{DV}_{n,\mu}$ and $\overline{\mathcal{DV}}_{n,\mu}$ are opposite; (c) membership of $\textsc{dyck}(\lambda)$ in the first \implies membership of $\textsc{dyck}(\lambda')$ in the second; (d) the injection $\iota_n$ maps each family into its corresponding family at level $n+1$; and (e) both family sizes tend to infinity as $n$ tends to infinity.

This is a proposed structural decomposition intended to explain the symmetry of the level-$k$ $q,t$-Catalan objects. The authors state that the conjectures are proved for $k ique 9$, while the general assertion remains open.