Approximately minimax detecting of a vector signal in Gaussian noise

In a normal vector sample $(X_1,\dots,X_N)^T$ of independent identically distributed variables $X_i\in\mathscr N(\xi,\Sigma)$, the сovarianсe matrix $\Sigma$ is not supposed to be known, and the hypothesis $H_0$: $\xi=0$ against $H_1$: $N\xi^T\Sigma^{-1}\xi=\delta$ is tested. The Hotelling test
$$ \Phi_N^0\colon T^2=N(N-1)X^TS^{-1}X>T_\varepsilon^2 $$
where
$$ \overline X=N^{-1}\sum_{i=1}^NX_i;\quad S=\sum_{i=1}^N(X_i-X)(X_i-X)^T $$
is proved to be approximately minimax for large samples in the following sense: for all (randomized) tests $\Phi$ of level $\alpha=\alpha_N$ under conditions
$$ O(\exp[-(\ln N)^{1/6}])\le\alpha\le1-O(\exp[-(\ln N)^{1/6}]) $$
and $\delta$'s under condition
$$ \exp[-(\ln N)^{1/6}]\le\delta\le(\ln N)^{1/6} $$
we have
$$ \sup_\Phi\inf_{\theta\in H_1}\mathbf E_\theta\Phi-\inf_{\theta\in H_1}\mathbf E_\theta\Phi_N^0=O_\varepsilon\biggl(\frac1{N^{i-\varepsilon}}\biggr) $$
for any $\varepsilon>0$.