Reflection-symmetric Dyck-path wreath conjecture

Let $k$ be a positive integer. Let $\mathbb{Z}_{2k+1}$ be the integers modulo $2k+1$, let $\mathcal{D}_{k}$ be the set of Dyck $k$-paths, and let $C_k$ denote the $k$th Catalan number. For a permutation $\pi$ of $\mathbb{Z}_{2k+1}$, let $\mathcal{F}_{\pi}$ be its associated wreath. For $D\in\mathcal{D}_k$, let $D^{(R)}$ be the Dyck path obtained by reflecting $D$ in the line $x=k$.

Reflection-symmetric strengthening. There exists a set $\Pi=\{\pi_1,\pi_2,\ldots,\pi_{C_k}\}$ of $C_k$ permutations of $\mathbb{Z}_{2k+1}$, each fixing $0$, and a bijection $\varphi:\Pi\to\mathcal{D}_k$ such that

$$\mathbb{Z}_{2k+1}^{(k)}=\bigcup_{i=1}^{C_k}\mathcal{F}_{\pi_i},$$

and, for any $i$ and $j$, the $j$th step of $\varphi(\pi_i)$ is a rise if and only if $\pi_i(j)\in\{1,2,\ldots,k\}$, and

$$\varphi^{-1}(D)(j)+\varphi^{-1}(D^{(R)})(-j)=0$$

for every Dyck $k$-path $D$ and every $j\in\mathbb{Z}_{2k+1}$.

This is the stronger statement suggested by the authors' search through $k\leq4$. It adds a reflection symmetry relating the permutations paired by reflected Dyck paths; the source gives no proof or resolution.