A refined squared cyclic ratio lower bound

Let $x,y,z>0$ ,prove that:

${\frac {{x}^{2}}{ \left( y+z \right) ^{2}}}+{\frac {{y}^{2}}{ \left( z+x \right) ^{2}}}+{\frac {{z}^{2}}{ \left( x+y \right) ^{2}}}\geq {\frac {1}{225}}\,{\frac { \left( 47\,{x}^{2}-22\,xy-22\,xz+47\,{y}^{2}-22\,yz+47\,{z}^{2} \right) \left( {x}^{2}+{y}^{2}+{z}^{2}+3\,xy+3\,yz+3\,xz \right) ^{2}}{ \left( y+z \right) ^{2} \left( z+x \right) ^{2} \left( x+y \right) ^{2}}}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_7252 (Apache-2.0). Complete source proposition preserved; proof developed independently.