The signed multiset-permutation and inversion-sequence descent–ascent equidistribution conjecture

Let $P^{\pm}(\{1,1,2,2,\ldots,n,n\})$ be the signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$, and let $\mathfrak I_{2n}^{(1,4,3,8,\ldots,2n-1,4n)}$ be the corresponding set of inversion sequences. Write $\operatorname{des}(\pi)$ for the descent statistic on signed multiset permutations and $\operatorname{asc}(\mathbf e)$ for the ascent statistic on inversion sequences. Signed multiset-permutation conjecture.

$$\sum_{\pi\in P^{\pm}(\{1,1,2,2,\ldots,n,n\})}x^{\operatorname{des}(\pi)}=\sum_{\mathbf e\in\mathfrak I_{2n}^{(1,4,3,8,\ldots,2n-1,4n)}}x^{\operatorname{asc}(\mathbf e)}.$$

This is an experimentally observed equality of descent and ascent distributions; the source gives no proof or resolution.