Lower-bounding Gaussian inner products with high probability

Question

Suppose that $K\subseteq \mathbb R^n$ is a proper convex set with piecewise smooth boundary and that $0 \in K$. Assume that $x \in K$ and let $z \sim \mathcal{N}(0, I_n)$ be a Gaussian random vector. The following holds with high probability when c < 1 and $n$ is sufficiently large: $$ c |\langle x, z\rangle| \leq \mathbb E \sup_{y \in K} |\langle y, z\rangle| $$ Now assume that $x$ lies in the boundary of $K$. Does there exist a similar result lower bounding $|\langle x, z\rangle|$ with high probability? I'm hoping for something like: with high probability, there exists $C > 1$ such that for $n$ sufficiently large, $$ C|\langle x , z\rangle| \geq \mathbb E \sup_{y \in K} |\langle y, z\rangle|. $$ Anything stronger would be a bonus! Thank you for your help.