Birational OY-invariant conjecture for the Lalanne–Kreweras involution

Let $\mathsf{A}^n$ be the type A poset and let $\mathcal{A}^{\mathrm{B}}(\mathsf{A}^n)$ be its positive birational labelings. For $pi\in\mathcal{A}^{\mathrm{B}}(\mathsf{A}^n)$, define the birational OY-invariant $\EuScript{Y}^{\mathrm{B}}$ by

$$\EuScript{Y}^{\mathrm{B}}(\pi)\coloneqq\prod_{[i,j]\in\mathsf{A}^n}\EuScript{Y}^{\mathrm{B}}_{[i,j]}(\pi),$$

where each factor is the product of the left and right ratios of maximal-chain weight sums specified in the preceding definition.

Birational OY-invariant conjecture. The function $\EuScript{Y}^{\mathrm{B}}\colon\mathcal{A}^{\mathrm{B}}(\mathsf{A}^n)\to\mathbb{R}_{>0}$ is invariant under both $\operatorname{Row}_{\mathcal{A}}^{\mathrm{B}}$ and $\operatorname{LK}^{\mathrm{B}}$. This would provide a birational invariant simultaneously for rowmotion and the birational Lalanne–Kreweras involution. The supplied excerpt gives the definition and an example but no resolution status, so the claim remains open.