Asymptotic equivalence of full and Gaussian maxT power

Let $\overline{H}$ be the matrix appearing in the maxT procedure, let $\iota$ denote the all-ones vector, and define

$$\overline{h}=\overline{H}\iota.$$

Suppose that $E$ has independent and identically distributed rows with distribution $\mathcal{N}(0,I)$, and that $\mu_l=\mu_1$ for every $1\leq l\leq p$. Let $Z_1\sim\mathcal{N}(n^{1/2}\mu_1,1)$, and let $Y$ have independent and identically distributed rows with distribution $\mathcal{N}(0,I)$. Writing $q_\alpha^{\overline{h}}$ and $q_\alpha^{\overline{h},Y}$ for the relevant maxT quantiles, asymptotic power equivalence conjecture. As $p,n\to\infty$,

$$\left|\frac{1}{p}\sum_{j=1}^p \mathbb{P}_E\left[n^{1/2}\mu_1+\iota'E_j>q_\alpha^{\overline{h}}\left(n^{1/2}\overline{h}'\iota\mu_1+\max_i\overline{h}'E_i\right)\right]-\mathbb{P}_{Z_1}\left[Z_1>q_\alpha^{\overline{h},Y}\left(n^{1/2}\overline{h}'\iota\mu_1+\max_{1\leq l\leq p}Y_l\right)\right]\right|\to0.$$

The conjecture asserts that the average rejection probability based on the full error matrix is asymptotically equivalent to the corresponding probability computed using an auxiliary Gaussian matrix. Its validity is presented heuristically in the paper, and no resolution is supplied in the source.