Particle-conjugation symmetry for multispecies inhomogeneous $t$-PushTASEP probabilities

Let $\mathcal{S}(\mathbf m)$ be the set of local-state configurations with multiplicity vector $\mathbf m$, and let $\widehat{\boldsymbol{\sigma}}_i$ denote the particle-conjugated local state obtained by reversing its species multiplicities. For a configuration $(\boldsymbol{\sigma}_1,\ldots,\boldsymbol{\sigma}_L)\in\mathcal{S}(\mathbf m)$, write $\mathbb{P}_l\!\left(\genfrac{}{}{0pt}{}{x_1,\ldots,x_L}{\boldsymbol{\sigma}_1,\ldots,\boldsymbol{\sigma}_L};t\right)$ for its probability, and set

$$\gamma=Lnl(l-1)/2-\sum_{0<j<n}j(n-j)m_j.$$

Particle-conjugation symmetry conjecture. The following equality holds:

$$\mathbb{P}_l\!\left(\genfrac{}{}{0pt}{}{x_1,\ldots,x_L}{\boldsymbol{\sigma}_1,\ldots,\boldsymbol{\sigma}_L};t\right)=(-1)^{n(n-1)/2}t^{\gamma}(x_1\cdots x_L)^{nl}\,\mathbb{P}_l\!\left(\genfrac{}{}{0pt}{}{x_1^{-1},\ldots,x_L^{-1}}{\widehat{\boldsymbol{\sigma}}_1,\ldots,\widehat{\boldsymbol{\sigma}}_L};t^{-1}\right).$$

Summing this proposed configuration-level symmetry over all configurations yields the particle-conjugation symmetry of the partition function stated in the preceding corollary. Establishing the stronger identity would therefore explain that corollary at the level of individual configurations.