Conjecture on the global-null correlation maximum bound

Let $f_n(x)$ be the density of $\sqrt{n}|S_{ij}|$, let $\bar{F}_n(x)=\int_x^{\sqrt{n}}f_n(x)$, and let $G_{n,p}(x)$ be the cumulative distribution function of the maximum of an independent $p\times p$ correlation matrix based on $n$ observed $p$-vectors, scaled by $\sqrt{n}$. Global-null correlation maximum conjecture. For fixed $k$,

$$\int_0^{\sqrt{\left(4-\frac{2}{k+2}\right)\log p}}G_{n,p}(x)\bar{F}_n^{k-1}(x)f_n(x)\,dx=o\left(\frac{1}{p^{2k}}\right).$$

This conjecture is the technical bound needed to apply the correlation-limit approximation when controlling the probability that $k$ independent correlations exceed the maximum correlation of a large correlation matrix. The paper states that it is needed to prove the asymptotic exponential law $T_k\stackrel{d}{\rightarrow}\operatorname{Exp}(1/k)$ under the global null, but reports that it has not yet been proved.