Bi-symmetric quadruple equidistribution on ascent sequences

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 the corresponding Euler–Stirling statistics. Bi-symmetric quadruple equidistribution. For each $n oindent\geq1$,

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

This is a conjectured strengthening of known symmetric distributions of ascent and repetition statistics on ascent sequences; it is supported by experimental data, while the stated equidistribution remains to be proved.