The TVK-shape sorting-probability conjecture

Fix an integer $d\ge 2$. Let $\alpha\in\mathbb R^d_{>0}$ be a Thoma sequence, let $\lambda\simeq \alpha n$ be a TVK $\alpha$-shape, and let $P_\lambda$ be the associated Young-diagram poset. Let $\delta(P_\lambda)$ denote its sorting probability.

TVK-shape sorting-probability conjecture. There is a universal constant $C>0$ such that

$$\delta(P_\lambda)\le \frac{C}{n^{5/4}}.$$

The paper proves an $O(n^{-1/2})$ upper bound in this setting and suggests that it is not tight. The conjectured $n^{-5/4}$ rate remains open.