a_j| \leq \hat{t}_j, \forall j) &=
    \mathbb{P}(\sqrt{n}|\hat\theta_j - \theta_j| \leq \hat{t}_j, \forall j;\
    \mathcal{E}_n) + 
    \mathbb{P}( \sqrt{n} |\hat\theta_j - \theta_j| \leq \hat{t}_j, \forall j;\
    \mathcal{E}_n^c)
 \\
& \leq
\mathbb{P}( \sqrt{n}|\hat\nu_j - \nu_j| \leq \hat{t}_j+\epsilon_n, \forall j) + 
\mathbb{P}(\mathcal{E}_n^c) \nonumber \\
& \leq
\mathbb{P}( |Z_{n,j}| \leq \hat{t}_j+\epsilon_n, \forall j) + 
C \frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6} + 
\frac{1}{n}
\\
& \leq \mathbb{P}( |Z_{n,j}| \leq \hat{t}_j, \forall j) + C \left[
\frac{\epsilon_n}{\underline{\sigma}} (\sqrt{2 \log b}  +2)  +
\frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6}  \right],\end{aligned}$$ where in the second-to-last inequality we have used the fact that $\mathbb{P}(\mathcal{E}^c_n) \leq \frac{1}{n}$ and in the last inequality we have applied the Gaussian anti-concentration inequality in (and have absorbed the term $\frac{1}{n}$ into higher order terms by increasing the value of $C$). A similar argument gives $$\mathbb{P}( \sqrt{n} |\hat\theta_j - \theta_j| \leq \hat{t}_j, \forall
    j)\geq \mathbb{P}( |Z_{n,j}| \leq \hat{t}_j, \forall j) - C \left[
\frac{\epsilon_n}{\underline{\sigma}} (\sqrt{2 \log b}  +2)  +
\frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6} \right].$$ To complete the proof, we will show that $$\label{eq:min}
\mathbb{P}( |Z_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall j) \geq
(1-\alpha) - \frac{1}{n} - \min \left\{ C \Delta_{3,n},  \frac{ \aleph_n z_{\alpha/(2s)}}{(\min_j
\gamma_j)^2} \left(\sqrt{ 2 + \log(2s ) } + 2 \right)\right\}.$$ Let $\hat{Z}_n \sim N(0,\hat \Gamma)$. By the Gaussian comparison , $$\mathbb{P}( |Z_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall j) \geq
\mathbb{P}( |\hat{Z}_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall j) - I
\geq 1 - \alpha - I$$ where $$I \leq 
\mathbb{E}\left[ \sup_{t = (t_1,\ldots,t_s) \in \mathbb{R}^s_+} \left|
    \mathbb{P}(|Z_{j,n