Archer et al.'s quasi-Stirling descent conjecture

Let $\mathcal{M}=\{1^2,2^2,\ldots,n^2\}$, and let $\overline{\mathcal{Q}}_{\mathcal{M}}$ denote the set of quasi-Stirling permutations of $\mathcal{M}$. For a permutation $\pi=\pi_1\pi_2\ldots\pi_{2n}$, define

$$des(\pi)=|\{i\mid \pi_i>\pi_{i+1},\ 1\leq i\leq 2n-1\}|+1.$$

Archer et al.'s conjecture. The number of $\pi\in\overline{\mathcal{Q}}_{\mathcal{M}}$ with $des(\pi)=n$ is equal to

$$(n+1)^{n-1}.$$

This conjecture concerns the descent enumerator of quasi-Stirling permutations on the multiset in which every element occurs twice. It refines the known enumeration $|\overline{\mathcal{Q}}_{\mathcal{M}}|=n!C_n$, where $C_n$ is the $n$th Catalan number; the supplied text gives no evidence that the descent-count assertion has been resolved.