[Here](https://math.stackexchange.com/a/3406957/631742), @MichaelRozenberg stated the following inequality without proof:

> 
> **Theorem.** For all non-negative $a,b,c\in\mathbb R$,
>  $$a^6+b^6+c^6-3a^2b^2c^2\geq16(a-b)^2(a-c)^2(b-c)^2.$$
> 
> 
> 

In my answer below I give a complete brute-force proof. However, more elegant proofs are welcome.