A linear upper bound for the Gaussian Benjamini–Hochberg false discovery rate

Let $\mathcal G^K_{\mathcal N}$ denote the class of Gaussian-dependent p-value vectors indexed by the hypotheses, and define

$$f_{K_0,K}(\alpha):=\sup_{\mathbf P\in\mathcal G^K_{\mathcal N}}\mathbb E\left[\frac{F_{\mathcal D_\alpha}}{R_{\mathcal D_\alpha}}\right],$$

where $K_0$ is the number of true null hypotheses. Let

$$f_K(\alpha):=\max_{K_0\in\mathcal K}f_{K_0,K}(\alpha).$$

Gaussian BH-FDR bound. There exists $c\geqslant 1$ such that

$$f_K(\alpha)\leqslant c\alpha$$

for all $\alpha\in(0,1)$ and $K\in\mathbb N$. This would give a dimension-free linear bound on the worst-case false discovery rate under Gaussian dependence. The paper states that the value of $c$, particularly at small significance levels, is important, but whether such a universal constant exists remains open.