Lower bound for the probability that a certain component of a Gaussian vector dominates all others

Question

Let $X\sim\cal N(\mu,\Sigma)$ be an $n$-dimensional Gaussian vector. I would like to estimate $$P(X_1>\max_{k=2,\dots,n}X_k).$$

While no closed form solution exists (see e.g. MO question on Maximal component of a multivariate Gaussian distribution), can one nevertheless obtain nontrivial lower bounds in the case that $\mu_1>\max_{k=2,\dots,n} \mu_k$?