Proving that $f(x_1,x_2\ldots x_n)=\frac{1}{n}\sum_{i}x_i^2 - \frac{1}{n^2}\sum_{i\neq j}\ln|x_i-x_j|\geq 4$

Question

I came across this inequality in Andersen, Giounnet and Zeitouni's book on Random Matrix Theory. Now, we are asked to establish this using the two inequalities $\ln|x_i-x_j|\leq \ln(|x_i|+1)+\ln(|x_j|+1)$ and $x^2-2\ln(1+|x|)\geq -4$. Further, we are required to verify that $-\frac{1}{n^2}\sum_{i\neq j}\ln|x_i-x_j|\leq f(x_1,x_2\ldots x_n)+4$. What I am able to verify directly is that $f(x_1,x_2\ldots x_n)\geq -4$, but not sure how to proceed. For context, this is involved in the proof that Dyson Brownian Motion never hits the boundary of the Weyl Chamber $x_1<x_2<x_3\ldots <x_n$. Any resources towards that end will also be helpful.