Schur's cubic sum bounds a product of squared differences

The best result here is the following.

Let $a$ , $b$ and $c$ be positive numbers. Prove that:

$a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)\geq\frac{4(a-b)^2(a-c)^2(b-c)^2}{a^3+b^3+c^3}$

Source: InternLM Lean-Workbook, record lean_workbook_20293 (Apache-2.0). Complete source proposition preserved; proof developed independently.