Conjectured chain decomposition of Dyck partitions

Fix $k$ and let $\lambda\in\operatorname{Par}(k)$. A Dyck partition is a partition equipped with the Dyck-partition structure used in the paper; $\operatorname{dp}_k(\operatorname{dyck}(\lambda))$ is the Dyck partition associated with $\lambda$. For a Dyck partition $\gamma$, $R(\gamma)=\{\nu^m(\gamma):0\leq m\leq\rho(\gamma)\}$ is its associated chain, and $\Delta$ and $\operatorname{defc}$ are the paper's statistics.

Conjectured chain decomposition. There exists a sequence of Dyck partitions $\{\gamma_{\lambda,i}:i\in\mathbb{Z}_{>0}\}$ with strictly increasing sizes satisfying all five stated conditions: it contains $\operatorname{dp}_k(\operatorname{dyck}(\lambda))$; its $\Delta$ values are weakly decreasing and then weakly increasing; all its elements have constant deficit statistic equal to

$$\sum_{r=1}^{\ell(\lambda)-1}(-1+\lambda_r)\sum_{s=r+1}^{\ell(\lambda)}\lambda_s;$$

for every $d\geq|\gamma_{\lambda,1}|$, exactly one element of the union of their chains has size $d$; and the corresponding unions for $\lambda$ and $\lambda'$ are $n$-opposite in every size $n$.

This conjecture is presented as a stronger structural generalization of the hook and almost-hook results. Its general validity is not established in the supplied text.