Quintuple equidistribution conjecture for Euler–Stirling statistics on permutations

Let $\mathsf{I}_n$ be the set of inversion sequences of length $n$, and let $\mathsf{asc}$, $\mathsf{dist}$, $\mathsf{zero}$, $\mathsf{\max}$ and $\mathsf{rmin}$ denote the corresponding inversion-sequence statistics. Define $\mathsf{rep}(s)=n-1-\mathsf{dist}(s)$, the number of repeated entries of $s$. Let $\mathfrak{S}_n$ be the symmetric group, and let $\Theta:\mathfrak{S}_n\to\mathsf{I}_n$ be a bijection satisfying the stated property (E:bv).

Quintuple equidistribution conjecture. There is a bijection $\Omega:\mathsf{I}_n\to\mathsf{I}_n$ such that, for every $s\in\mathsf{I}_n$,

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

Consequently, for every $\pi\in\mathfrak{S}_n$,

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

The conjecture concerns a quintuple equidistribution of Euler–Stirling statistics on permutations and is stated as remaining unsolved in the source.