$\max_{1 \leq i \leq n}|\langle x,w_i\rangle|$, $\max_{1 \leq i \leq n}|\langle y,w_i\rangle|$ at same $w_i$ if $x$ and $y$ are close enough?

Question

Let $x,y \in \mathbb{C}^n$ with $|x|_2 = |y|_2 = 1$. Let $w_1, \ldots, w_N \in \mathbb{C}^n$. Let $j,k \in \{1,\ldots, N\}$ such that $$ |\langle x,w_j\rangle| = \max_{1 \leq i \leq n}|\langle x,w_i\rangle| \quad \text{and} \quad |\langle y,w_k\rangle| = \max_{1 \leq i \leq n}|\langle y,w_i\rangle|. $$ I think I should be able to force $w_j=w_k$ by assuming $|x-y|_2$ is small enough, but I'm having trouble justifying this.

Can someone help me out?