I am interested in computing $$\lim_{x \to \infty} x \Phi_n(\mu_x; 0,\Sigma_x),$$ where $\Phi_n(\cdot)$ is the cumulative distribution function of a multivariate normal of dimension $n$ evaluated in $\mu_x$, centered in $0$ and with covariance matrix $\Sigma_x$. Here, $\mu_x=(\mu_x, \dots, \mu_x)$ is a constant vector with $$\mu_x \sim -\sqrt{-2\log (a/x) -\log(-2\log (a/x)) - \log(2\pi)},$$ for $x \to \infty$. The value $a>0$, is a fixed real parameter and also $n$, the dimension of the multivariate Normal, is fixed. The covariance matrix $\Sigma_x$ is in correlation form with a common correlation coefficient $\rho= 2\log(x)/(1+2\log(x))$. From numerical simulations, it seems $\lim_{x \to \infty} \Phi_n(\mu_x; 0,\Sigma_x) \approx a/(xn)$, hence the initial limit converges to $a/n$. I want to show it analytically. Specifically, I am interested in computing the exact value of the limit, not only the convergence.
Are there any asymptotic approximations to evaluate the multivariate Normal distribution function? I tried with the multivariate Mill's ratio (https://nvlpubs.nist.gov/nistpubs/jres/66B/jresv66Bn3p93_A1b.pdf), and the limit converges, but the specific value to with it converges is greater than 1, so doesn't make any sense. I think this depends on the facts that the correlations are close to one.