The probability that the minimum of a multivariate Gaussian exceeds zero

Suppose $X sim mathcal N_n(text{diag}(Sigma), sigma^2 Sigma)$ where $Sigma$ may be allowed to be low rank, and let $Y = min_i X_i$. What can be said about $Pleft(Y geq 0right)$?

In general I know that the exact distributions of Gaussian order statistics can be intractable, such as this math.se Q&A and the discussion here, but I’m hoping that the relationship between the mean and covariance matrix may lead to some simplification, or how I don’t need the distribution of $Y$ but rather just the probability that it is greater than zero. The $X_i$ not being iid prevents me from using the usual things I know for examining minima and maxima but I’m still hoping something can be done aside from numerical integration/simulation given values of $Sigma$ and $sigma$. I’d be very interested in approximations too.

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The context on this and the unusual mean vector come from a now-deleted question on stats.se that essentially asked the following: if we have $Xsimmathcal N_k(mathbf 0, sigma^2 I)$ and nonrandom nonzero vectors $z_1,dots,z_ninmathbb R^k$, what is the probability that $|X|^2 leq |X-z_i|^2$ for all $i$?

$|X-z_i|^2 = |X|^2 – 2 X^Tz_i + |z_i|^2$ so the question is equivalent to $P(|z_i|^2- 2 X^Tz_i geq 0 text{ for all }i)$. I collected the $z_i$ into the columns of a $ktimes n$ matrix $Z$ so I can write the random variables in question as an affine transformation of $X$ via
$$
text{diag}(Z^TZ) – 2 Z^TX sim mathcal N_n(text{diag}(Z^TZ), 4sigma^2 Z^TZ)
$$
and I want the probability that this random vector is all non-negative, so this led me to the question I asked. The factored form of $Sigma$ here is why I want to allow for possibly low rank covariance matrices since I could have $k leq n$.