A linear upper bound for the Gaussian Simes-test error probability

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

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

Gaussian Simes bound. There exists $c\geqslant 1$ such that

$$s_K(\alpha)\leqslant c\alpha$$

for all $\alpha\in(0,1)$ and $K\in\mathbb N$. If true, determining the smallest possible $c$, especially for practically relevant small values of $\alpha$, is important. The paper notes that the currently available general bound grows with $K$, so whether a universal constant exists remains open.