A squared pairwise ratio sum with a difference-product correction

Prove that ${\frac { \left( y+z \right) ^{2}}{{x}^{2}+yz}}+{\frac { \left( z+x \right) ^{2}}{{y}^{2}+zx}}+{\frac { \left( x+y \right) ^{2}}{{z}^{2}+xy}}\geq 6+5\,{\frac { \left( y-z \right) ^{2} \left( z-x \right) ^{2} \left( x-y \right) ^{2}}{ \left( {x}^{2}+yz \right) \left( {y}^{2}+zx \right) \left( {z}^{2}+xy \right) }}$ for $x,y,z>0$

Source: InternLM Lean-Workbook, record lean_workbook_36275 (Apache-2.0). Complete source proposition preserved; proof developed independently.