Finite-sample consistency conjecture for empirical kernel quantile embeddings with p greater than 1

Let $\mathcal{X} \subseteq \mathbb{R}^d$, let $\nu$ have a density, and let $P$ and $Q$ be measures on $\mathcal{X}$ with densities bounded away from zero, satisfying $f_P(x) \geq c_P>0$ and $f_P(x) \geq c_Q>0$. Suppose

$$\mathbb{E}_{X \sim P} [k(X, X)^{\nicefrac{p}{2}}]<\infty, \qquad \mathbb{E}_{X \sim Q} [ k(X, X)^{\nicefrac{p}{2}}]<\infty,$$

for samples $x_{1:n} \sim P$ and $y_{1:n} \sim Q$. Finite-sample consistency conjecture.

$$\mathbb{E}_{\substack{x_{1:n}\sim P \\ y_{1:n}\sim Q}}\left| \operatorname{e-KQD}_p(P_n, Q_n;\nu, \gamma_l) - \operatorname{e-KQD}_p(P, Q;\nu, \gamma) \right| =\mathcal{O}(l^{-\nicefrac{1}{2}} + n^{-\nicefrac{1}{2}}).$$

This conjecture predicts an $n^{-1/2}$-type finite-sample convergence rate for empirical kernel quantile divergences when $p>1$, extending the established $p=1$ argument. The preceding Wasserstein convergence result shows that such a rate requires suitable integrability conditions, and the conjecture remains unproved in the source.