Probability of a gaussian lying in half-spaces?

Question

Let ${\cal N}(0,I)$ denote a Gaussian distribution.

Let $W \in \mathbb{R}^{d\times k}$ denote a rank-$k$ matrix with $d \geq k$.

Let $S$ denote $\{ x \in \mathbb{R}^d ~|~ W^\top x \geq 0 \}$.

What is the lower bound of $ \underset{x \sim {\cal N}(0,I)}{Pr} [ x \in S]$ in terms of parameters $k,d,\sigma_\min(W), \sigma_\max(W)$?