A cyclic cubic product sum is at most two hundred fifty-six twenty-sevenths at fixed total four

Let $a$ , $b$ , $c$ and $d$ positive real numbers such that: $a+b+c+d=4$ . Prove that: $a^2b+b^2c+c^2d+d^2a \leq \frac{256}{27}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_81777 (Apache-2.0). Complete source proposition preserved; proof developed independently.