The order of the Kolmogorov-Smirnov distance for the bootstrap of $U$-quantiles is considered. We observe that the order of the bootstrap of $U$-quantiles depends on the order of the local variance of the first term of the Hoeffding decomposition at the $U$-quantile. This order can be smaller than the order of the bootstrap of quantiles: $U$-quantiles can be smoother than quantiles.
"The Asymptotic Accuracy of the Bootstrap of $U$-Quantiles." Ann. Statist. 23 (5) 1802 - 1822, October, 1995. https://doi.org/10.1214/aos/1176324324