Fix some $q\in\mathbb N$ and some probability $p\in[0,1]$. Denote with $F_n$ the  the q-th highest oder statistic (i.e. the distribution of the q-th highest draw) of $n$ draws from a uniform distribution on $[0,1]$. Obviously, for $n\to\infty$, the $p$-quantile of $F_n$ as well as the expectation of the lower $p$-quantile approach $1$. I am interested in the speed they converge to each other. More precisely, I need to determine

$ \lim_{n\to\infty} n\big(F_n^{-1}(p)-\mathbb E_{F_n}[X|X\leq F_n^{-1}(p)]\big)$

It seems related to the question https://mathoverflow.net/questions/346157/the-behavior-of-a-uniform-order-statistic-near-zero, but I don't see how I can solve it.