Global chain decomposition conjecture for $q,t$-Catalan numbers

Let $u$ be the partition map whose iterates define the $u$-segments, let $u^*(\gamma)$ denote the set of all defined iterates $u^m(\gamma)$, and let $\operatorname{Par}$ be the set of partitions. For each deficit partition $\mu$, write $\operatorname{Def}(k)$ for the set of partitions of deficit $k$, $\operatorname{defc}(\gamma)$ for the deficit of $\gamma$, $|\mu|$ for the size of $\mu$, $\ell(\mu)$ for its length, and $\operatorname{tail}(\mu)$ for its $\nu$-tail. For a collection $\mathcal{S}$ of partitions, define

$$\operatorname{Cat}_{n,\mathcal{S}}(q,t)=\sum_{\gamma\in\mathcal{S}\cap\mathcal{DP}(n)}q^{\operatorname{area}_n(\gamma)}t^{\operatorname{dinv}(\gamma)}.$$

Global chain decomposition conjecture. There exist collections of partitions $\mathcal{C}_{\mu}$, indexed by deficit partitions $\mu$, and a size-preserving involution $\mu\mapsto\mu^*$ on $\operatorname{Par}$, satisfying: the collections $\mathcal{C}_{\mu}$ are pairwise disjoint; every $\gamma\in\mathcal{C}_{\mu}$ has $\operatorname{defc}(\gamma)=|\mu|$; each collection has the form $\{C_{\mu}(a),C_{\mu}(a+1),C_{\mu}(a+2),\ldots\}$, where $a=\ell(\mu^*)$ and $\operatorname{dinv}(C_{\mu}(i))=i$ for all $i\geq a$; every $\gamma\in\operatorname{Def}(k)$ belongs to $\mathcal{C}_{\mu}$ for some $\mu\in\operatorname{Par}(k)$; $\operatorname{tail}(\mu)\subseteq\mathcal{C}_{\mu}$ for all $\mu$; if $\gamma\in\mathcal{C}_{\mu}$ and $\nu(\gamma)$ is defined, then $\nu(\gamma)\in\mathcal{C}_{\mu}$; and, for all $n\geq 0$ and all $\mu$,

$$\operatorname{Cat}_{n,\mathcal{C}_{\mu}}(q,t)=\operatorname{Cat}_{n,\mathcal{C}_{\mu^*}}(t,q).$$

This conjecture is the global-chain version of the structural conjecture attributed to parts (a), (b), (c), (d), (f), (g), and (j) of Conjecture 6.9 of LLL18. The paper states that its local-chain conjecture implies this global version; the existence of the required collections and involution is therefore the principal unresolved structural assertion here.