Dyck-path strengthening of the 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.

Dyck-path 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 $\pi_i(j)\in\{1,2,\ldots,k\}$.

For $n=2k+1$, the original wreath conjecture requires $C_k$ wreaths, and this statement strengthens it by indexing the wreaths with Dyck paths and prescribing the rise/value correspondence. The source reports computer verification for $k\leq4$, but no general proof.