Shift-invariance conjecture for stochastic colored vertex-model probabilities

Let $n$ be a positive integer, let $\alpha,\delta\in[1,n]$, and let $w\in S_n$. Set $\beta:=w^{-1}(\delta)$. For indices $\alpha,\delta$, let $\gamma:=w(\alpha)$; the pair $\\{\alpha,\beta\\}$ is an inversion of $w$ if either $\alpha<\beta$ and $\gamma>\delta$, or $\alpha>\beta$ and $\gamma<\delta$. For any sets $\mathcal{H}$ and $\mathcal{V}$, write $\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})$ for the corresponding probabilities in the stochastic colored vertex model, with the displayed boundary configurations determining the events.

Shift-invariance conjecture. If $\\{\alpha,\beta\\}$ is not an inversion of $w$, then the three probability ratios satisfy

$$\frac{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{upper-left}}}{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{corner}}}=\left.\frac{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{lower-right}}}{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{corner}}}\right|_{z_\alpha\leftrightarrow z_\beta},\qquad \frac{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{central}}}{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{corner}}}=\left.\frac{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{central}}}{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{corner}}}\right|_{z_\alpha\leftrightarrow z_\beta},$$ $$\frac{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{lower-left}}}{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{corner}}}=\left.\frac{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{upper-right}}}{\mathbf{P}^{\mathcal{H},\mathcal{V}}(\mathbf{x},\mathbf{y})_{\text{corner}}}\right|_{z_\alpha\leftrightarrow z_\beta}.$$

Here $z_\alpha\leftrightarrow z_\beta$ denotes exchanging the two spectral parameters.

The conjecture concerns shift-invariance identities for probabilities associated with the distribution $Y^w$; it was verified computationally for $n\leq 6$. The supplied text also gives a counterexample when $\\{\alpha,\beta\\}$ is an inversion, so the conjecture as stated is refuted.