Coherence is a long term notion in the literature of multiple testing. From FWER control multiple testing theory to confidence envelopes on the number of false positives, coherence is a desirable property for multiple testing procedures. However, we show in this article that coherence is not a sufficient property to ensure that a multiple testing procedure is nonimprovable via an interpolation mechanism. We introduce a new no- tion, the notion of strong coherence, which is a stronger property than coherence and ensures that a multiple testing procedure is nonimprovable via the interpolation mech- anisms considered in the literature. Finally, we provide a uniform improvement of the confidence envelope from Durand et al. (2020) that is computable in polynomial time and demonstrate the improvement on simulated data.

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