A quadratic pair-sum ratio bounds a squared square-root sum

If $x$ , $y$ , $z$ are positive reals, prove that $\frac{\left(y+z\right)^{2}}{y+z+2x}+\frac{\left(z+x\right)^{2}}{z+x+2y}+\frac{\left(x+y\right)^{2}}{x+y+2z}\geq\frac{\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)^{2}}{3}$.

Source: InternLM Lean-Workbook, record lean_workbook_5580 (Apache-2.0). Complete source proposition preserved; proof developed independently.