Equality of the Gaussian worst-case FDR and Simes error bounds

Let $\mathcal G^K$ be the class of Gaussian-dependent vectors of $K$ p-values, and let $S_K(\mathbf P)$ be the Simes p-value. Define

$$s_K(\alpha):=\sup_{\mathbf P\in\mathcal G^K}\mathbb P\bigl(S_K(\mathbf P)\leqslant\alpha\bigr).$$

For the Benjamini–Hochberg procedure at level $\alpha$, let $f_K(\alpha)$ be the maximum false discovery rate over the possible numbers of true null hypotheses and Gaussian-dependent p-value vectors. Gaussian FDR–Simes equality.

$$f_K(\alpha)=s_K(\alpha)$$

for all $\alpha\in(0,1)$ and $K\in\mathbb N$. The equality would identify the worst-case Gaussian FDR with the worst-case Gaussian Simes-test error probability. The paper explicitly says that it remains unclear whether these quantities are equal, so the claim is open.