The no-positive-equal-coordinates conjecture for z-measures on the Thoma simplex

Let $\mathcal M_{z,z';\theta}$ be the boundary z-measure on the Thoma simplex $\Omega$, whose points are written $(\alpha;\beta)$ with nonincreasing nonnegative coordinate sequences $(\alpha_i)$ and $(\beta_i)$. No-positive-equal-coordinates conjecture. For every $i\geq 1$, the sets

$$\{(\alpha;\beta)\in\Omega:\alpha_i=\alpha_{i+1}>0\},\qquad \{(\alpha;\beta)\in\Omega:\beta_i=\beta_{i+1}>0\}$$

are null sets with respect to $\mathcal M_{z,z';\theta}$. This is the second conjectural property assumed for the paper's main results; the supplied text gives no resolution status.