Recently I have come across a ranked version of Dirichlet distribution whose density on the open simplex $\Delta^{n-1}_+$ is given by $$ p(\vec x) = B^{-1} \prod_{k=1}^n x_{(k)}^{\alpha_k -1},\quad \vec x \in \Delta^{n-1}_+, $$ where $B = \int_{\Delta^{n-1}} \prod_{k=1}^n x_{(k)}^{\alpha_k -1} dx$ is the normalizing factor, $x_{(k)}$ is the $k-$th largest among $x_1, x_2,\cdots, x_n$ and $\alpha_k$'s positive. (The only difference from the usual Dirichlet distribution is $x_k$ being replaced by the rank-based $x_{(k)}$ in pdf.)

We are interested in under what conditions on $\alpha_k$'s will make the following statement holds true: Assume $X^{n}$ follows the ranked Dirichlet distribution in dimension $n$ with parameters $\{\alpha^n_k\}_{k=1}^n$, for any fixed small $\epsilon >0$, $$ \sup_{n} \left\{\frac{1}{n^2}\sum_{k=1}^{n-1} \sum_{l>k} \mathbb{P} \left( X^n_{(k)} - X^n_{(l)} < \frac{\epsilon}{n}\right) \right\} \leq f(\epsilon) $$ for some function $f$ such that $\lim_{\epsilon \rightarrow 0} f(\epsilon) = 0$.

Intuitively, this says the averaging probability of gaps between coordinates being less than a dimension-scaled threshold is small. While it seems a pure calculus problem, the computation of the target probability becomes tricky once involved with this "rank" structure. I greatly appreciate any suggestions on how to bound this probability with some choice of $\alpha$ or any reference where you may have seen similar stuff.