Asymptotic linear-form conjecture for empirical transport-based quantiles

Let $P_n$ be the empirical measure based on independent observations $X_1,\ldots,X_n$ from $P$, let $z$ lie in the setting of Theorem InfluenceQuantiles-main, and let $\mathbf{I}(X_i;\mathbf{Q}_P(z))$ denote the influence function. Define

$$r_n=\begin{cases} \sqrt{n/\log(n)},&d=2,\\\\ n^{1/d},&d\geq 3. \end{cases}$$

Asymptotic linear-form conjecture. Under the setting of Theorem InfluenceQuantiles-main,

$$r_n\bigl(\mathbf{Q}_{P_n}(z)-\mathbf{Q}_P(z)\bigr)=\frac{r_n}{n}\sum_{i=1}^n\mathbf{I}(X_i;\mathbf{Q}_P(z))+o_{\mathbb{P}}(1).$$

The conjecture proposes a nonstandard convergence rate $r_n^{-1}$, reflecting the singular, non-square-integrable influence function and the resulting heavy-tailed behavior; numerical evidence supports the proposed scaling, while the pointwise vector-valued linearization remains unproved.