Central-measure support conjecture for the pascalized coupled Young graph

Let $\Theta$ be the coupled Young graph, let $\mathcal{P}(\Theta)$ be its pascalization, and let $A_{(\mathcal{C}_H,\delta)}(\infty)$ be the associated inductive limit algebra at the generic parameter. A central measure is a central measure on the branching graph $\mathcal{P}(\Theta)$. Coupled Young boundary-reduction conjecture. Any central measure on $\mathcal{P}(\Theta)$ associated with $A_{(\mathcal{C}_H,\delta)}(\infty)$ at the generic parameter is fully supported on $\Theta$, and therefore, by the established boundary result for $\Theta$, on the Young graph $\mathbb{Y}\subset\Theta$. If true, the boundary of $\mathcal{P}(\Theta)$ would reduce to the Thoma simplex. The paper states that this remains open and seeks to derive it from a numerical conjecture.