Let $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ be two unit vectors such that $\sum_{i}{x_i}=\sum_{i}{y_i}=0$ and

$$x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n} \gt 1-\frac{1}{n}.$$

Can we prove that $$x_{1}y_{\sigma(1)}+x_{2}y_{\sigma(2)}+\cdots +x_{n}y_{\sigma(n)} \neq 0$$

for all permutations $\sigma$?