Power improvement of BH via randomization under PRDS

Determine whether randomization can strictly improve the power of the Benjamini–Hochberg (BH) procedure under positive regression dependence on a subset (PRDS). Specifically, show whether there exists any randomized multiple testing procedure that strictly dominates BH under PRDS, or prove that BH is uniformly undominated by randomized procedures in this dependence regime.

Background

The book discusses randomized combination rules that strictly improve certain e-value based procedures, and then contrasts this with classical p-value based procedures. The authors explicitly state that they do not know of any way for randomization to improve BH under PRDS, and they conjecture BH cannot be strictly improved in any situation without worsening power elsewhere when only PRDS-type assumptions are made.

This frames a precise open question about the role of randomization in multiple testing under PRDS: either find a randomized procedure that strictly dominates BH, or establish that BH is uniformly undominated by randomized methods under PRDS.