TSTST 2012 Problem 6

Positive real numbers $x, y, z$ satisfy $x y z+x y+y z+z x=x+y+z+1$. Prove that $$\frac{1}{3}\left(\sqrt{\frac{1+x^{2}}{1+x}}+\sqrt{\frac{1+y^{2}}{1+y}}+\sqrt{\frac{1+z^{2}}{1+z}}\right) \leq\left(\frac{x+y+z}{3}\right)^{5 / 8} .$$