OSN 2021 Problem 4

Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$ . Prove that $$2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x}$$