Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$ and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does $ \min_{1\le i\ne j\le d}\left(x_{i}+x_{j}-\sqrt{x_{i}^{2}+x_{j}^{2}+2x_{i}x_{j}\cos\left(y_{i}-y_{j}\right)}\right)\le\frac{80}{d^{3}} $ hold?
Thanks for any helpful answers.