Quadruple equidistribution conjecture for ascent-sequence statistics

Let $\mathcal{A}_n$ be the set of ascent sequences of length $n$, and let $\mathsf{asc}$, $\mathsf{rep}$, $\mathsf{zero}$ and $\mathsf{\max}$ denote their ascent-sequence statistics.

Quadruple equidistribution conjecture. For every $n\geq1$,

$$\sum_{s\in\mathcal{A}_n}u^{\mathsf{asc}(s)}x^{\mathsf{rep}(s)}z^{\mathsf{zero}(s)}q^{\mathsf{\max}(s)}=\sum_{s\in\mathcal{A}_n}u^{\mathsf{rep}(s)}x^{\mathsf{asc}(s)}z^{\mathsf{\max}(s)}q^{\mathsf{zero}(s)}.$$

Equivalently, the statistic quadruples $(\mathsf{asc},\mathsf{rep},\mathsf{zero},\mathsf{\max})$ and $(\mathsf{rep},\mathsf{asc},\mathsf{\max},\mathsf{zero})$ are equidistributed on ascent sequences. The source states that this conjecture is affirmed in the paper, so it is no longer open.