Empirical-process convergence conjecture for pseudo-observations

Let $F$ be the distribution function of the transformed observations, let $\hat F_n$ be the empirical distribution function based on the rank-based pseudo-observations, and define

$$B_n:=\sqrt{n}(\hat F_n-F).$$

Let $W$ be a standard Wiener process and let $V$ be a stochastic process on $[0,1]$. Empirical-process convergence conjecture. As $n\to\infty$, $B_n$ converges weakly in $L^\infty([0,1])$ to

$$B:=W\circ F+V,$$

where the sample paths of $V$ are almost surely locally Lipschitz continuous at $x$. Moreover, on a possibly enriched probability space containing the data and $B$, there exists $\alpha>0$ such that

$$\lVert B_n-B\rVert_\infty=\operatorname{O}_p\mleft(n^{-\frac{1+\alpha}{10}}\mright).$$

This high-level conjecture is introduced to analyze the asymptotic behavior of the kernel density estimator based on dependent rank-based pseudo-observations. The weak convergence and coupling-rate assertion provide the empirical-process input needed for that analysis; the source gives no evidence that the conjecture has been resolved.