The staircase Young tableaux–sorting networks generating-function identity

For $n\ge 2$, let $\delta_n$ be the staircase Young diagram, let $\operatorname{SYT}(\delta_n)$ be its standard Young tableaux, and let $\operatorname{SN}_n$ be the set of sorting networks on $n$ elements. For each $t\in\operatorname{SYT}(\delta_n)$ and $s\in\operatorname{SN}_n$, let $f_t(x_1,\ldots,x_{n-1})$ and $g_s(x_1,\ldots,x_{n-1})$ be the associated generating polynomials, and let $\sigma_t$ and $\pi_s$ denote the corresponding basis vectors. The staircase tableaux–sorting networks conjecture. For $n\ge 2$, one has the identity of vector-valued generating functions

$$\sum_{t\in\operatorname{SYT}(\delta_n)} f_t(x_1,\ldots,x_{n-1})\,\sigma_t = \sum_{s\in\operatorname{SN}_n} g_s(x_1,\ldots,x_{n-1})\,\pi_s.$$

This is presented as an algebraic-combinatorial reformulation of the preceding distributional conjecture and is intended to clarify the relation between staircase Young tableaux and sorting networks. The supplied passage does not state a resolution, so the identity remains open.