nested sampling X check

Andrew Fowlie, Will Handley and Liangliang Su have recently arXived a new paper on checking the convergence of nested sampling by a uniformity test. The argument goes as follows: if the draw from the prior under the likelihood restriction (at the core of the nested sampling principle) is correctly generated, the rank of the realised value of the associated likelihood should be uniformly distributed among the remaining likelihoods. Obviously, the opposite does not hold:  a perfectly uniform distribution can happen even when the sampler misses a particularly well-hidden mode of the target disstribution or when it systematically stops too early, using for instance a misspecified bound on the likelihood. One particular setting when uniformity fails is when the likelihood surface plateaus in a particular region of the parameter space. (As a French speaker, writing plateaus makes me cringe since the plural of plateau is plateaux! Pardon my French!) When reaching the plateau the algorithm starts accumulating at the limiting value (or else completely ignores the plateau and its prior mass). I actually wonder if the existence of plateaux is not a sufficient reason for invalidating nested sampling, at least in its original version, since it assumes a continuous distribution on the likelihood values… If no plateau comes to hinder the algorithm, the rank test could be used to calibrate the exploration algorithm as for instance in the determination of the number of MCMC steps, running in parallel T random walks until the rank test across these runs turns green. The authors of the paper suggest using a Kolmogorov-Smirnov test, which strikes me as not the most appropriate solution, given the discrete nature of the theoretical distribution and the existence of uniformity tests in the pseudo random generation literature. At a conceptual level, I am also wondering at the sequential use of the test (as opposed to a parallel version at each iteration) since the target distribution is changing at every step (and so does the approximate method used to reproduce the prior simulation under the likelihood restriction).