Joint probability for nearly independent Gaussians

Question

Suppose that $X_1, \dots, X_n$ are jointly Gaussian and are nearly independent. What I mean by this is that the covariance matrix $\Sigma$ is given by $$\Sigma_{ij} = \begin{cases}1 & i = j\\\epsilon_{ij} & i \ne j\end{cases}$$ where the $\epsilon_{ij} = \epsilon_{ji}$ are bounded by some small constant $\epsilon$ in absolute value, and have random (Rademacher) sign. If all of the $\epsilon_{ij}$ were zero, we could say that $$\mathrm{Pr}\left[X_n \le c \mid \bigcap_{i=1}^{n-1}(X_i \le c)\right] = \mathrm{Pr}[X_n \le c] = \Phi(c)$$ where $\Phi$ is the CDF for the standard normal distribution. What I would like to know is, given the above covariance for the $X_i$, can we say that $$\mathrm{Pr}\left[X_n \le c \mid \bigcap_{i=1}^{n-1}(X_i \le c)\right] \approx \Phi(c)$$ That is, can we find some function $f(\epsilon)$ such that $$\left | \mathrm{Pr}\left[X_n \le c \mid \bigcap_{i=1}^{n-1}(X_i \le c)\right] - \Phi(c)\right | \le f(\epsilon)$$ An asymptotic bound is also acceptable.