Symmetric equidistribution conjecture for inversion-sequence statistics

Let $\mathcal{I}_n$ be the set of inversion sequences of length $n$, let $\mathfrak{S}_n$ be the symmetric group, and let $b:\mathfrak{S}_n\to\mathcal{I}_n$ be the bijection due to Baril and Vajnovszki. Let $\mathsf{asc}$, $\mathsf{rep}$, $\mathsf{zero}$, $\mathsf{\max}$, and $\mathsf{rmin}$ be statistics on inversion sequences, and let $\mathsf{des}$, $\mathsf{iasc}$, $\mathsf{lmax}$, $\mathsf{lmin}$, and $\mathsf{rmax}$ be the corresponding statistics on permutations. Symmetric equidistribution conjecture. There is a bijection $mathit{\Omega}:\mathcal{I}_n\to\mathcal{I}_n$ such that, for all $s\in\mathcal{I}_n$,

$$(\mathsf{asc},\mathsf{rep},\mathsf{zero},\mathsf{\max},\mathsf{rmin})s =(\mathsf{asc},\mathsf{rep},\mathsf{zero},\mathsf{rmin},\mathsf{\max})\Omega(s).$$

Consequently, for all $mathit{\pi}\inmathfrak{S}_n$,

$$(\mathsf{des},\mathsf{iasc},\mathsf{lmax},\mathsf{lmin},\mathsf{rmax})\pi =(\mathsf{des},\mathsf{iasc},\mathsf{lmax},\mathsf{rmax},\mathsf{lmin})(b^{-1}\circ\Omega\circ b)(\pi).$$

The conjecture is the inversion-sequence analogue of the established symmetric equidistribution for ascent sequences. Its permutation formulation follows formally from the asserted bijection and the Baril–Vajnovszki bijection, but the existence of $mathit{\Omega}$ remains open in the supplied source.