Mirror and knockoff+ thresholds under dependence

Many multiple-testing methods compare the two sides of a null distribution to control the false discovery rate (FDR). Small $p$-values or large positive scores are treated as possible discoveries, while large $p$-values or large negative scores are used to estimate how many of those discoveries are false. The mirror and knockoff+ thresholds are built on this idea. For valid knockoff statistics, the comparison is justified by a strong property: conditional on their magnitudes and the nonnull scores, the null signs are independent fair coin flips.
This paper asks what can go wrong when the same threshold is used without that property. We give three answers. First, we construct exactly uniform $p$-values satisfying positive regression dependence on a subset (PRDS), with a joint density that is positive throughout the unit cube. At a nominal level of 10%, an example with eleven hypotheses has FDR 17.4%, and within the same family the FDR can approach one half. Second, for standard Gaussian null scores with any fixed positive equicorrelation, however small, the FDR eventually exceeds every level below one half as the number of hypotheses grows. Third, when null scores may have different scales, a positive-definite Gaussian model can make the FDR arbitrarily close to one. Numerical experiments show that these failures are visible at moderate dimensions.
These results do not contradict knockoff theory. They show that the shared counting rule is not, by itself, an FDR guarantee: validity depends on the joint behavior of the null signs, not only on marginal symmetry, Gaussianity, or PRDS.