Here the black is the population quantity that you are trying to estimate. How is it defined in population terms. It seems like it is the expecctation of some quantity conditioned on an order statistics from some training sample, but I can't figure out what it is more precisely.
My intuition for this problem is built around estimation of a survival function, which may be far enough from your problem to be misleading, but, following that analogy, Iwould expect that you won't do much better pointwise, as the monitonicity forces pointwise coverage to hold uniformly as well. Isouldn't expect that you'll get a much more narrow interval pointwise.