Uniform convergence conjecture for the limiting rook-placement profile

Let $\lambda\in\mathcal{D}_n$ be a fixed partition, and let $\varepsilon>0$. For a uniformly random rook placement $\pi\sim\operatorname{Unif}(\operatorname{RP}(N\odot\lambda))$, write $\widetilde{\xi}_{\pi}(t)$ for its rescaled profile and $\mathfrak{m}_{\lambda}(t)$ for the limiting mean profile. Uniform convergence conjecture.

$$\mathbb{P}\!\left(\sup_{t\in[0,2]}\left|\widetilde{\xi}_{\pi}(t)-\mathfrak{m}_{\lambda}(t)\right|<\varepsilon\right)\longrightarrow 1$$

as $N\to\infty$. This predicts convergence in probability, uniformly over $t\in[0,2]$, of the rescaled rook-placement profile to its deterministic limiting mean. The preceding proposition establishes convergence of expectations at each fixed $t$, while the conjecture strengthens this to uniform probabilistic convergence.