Comparison of critical regions of Multiple univariates and multivariate tests.

Question

Let $X_i\sim\mathcal{N}(\mu_i,\sigma_i^2)$ for $i = 1,\ldots,n$. Also let $\boldsymbol{X} = (X_1,\ldots,X_n)^{\rm T}\sim MVN(\boldsymbol{\mu},\mathrm{\Sigma})$, where $\boldsymbol{\mu} = (\mu_1,\ldots,\mu_n)^{\rm T}$. I am expecting the following inequality to be true: $\mathbb{P}(|\sqrt{n}(\overline{X}_i-\mu_i)/\sigma_i|\leq \alpha:\;\mbox{for all}\;i=1,\ldots,n)\geq \mathbb{P}(n(\boldsymbol{X}-\boldsymbol{\mu})^{\rm T}\mathrm{\Sigma}^{-1}(\boldsymbol{X}-\boldsymbol{\mu})\leq \alpha^2)$. It can be proved for $n=2$ by comparing the regions inside the probabilities (first one is rectangle and the second is ellipse). Any hint to to prove it for any $n$?