A sixth-degree cyclic polynomial inequality

Let $x,y,z>0,$ $\left( {z}^{4}+x{y}^{3}+2\,x{z}^{3}+{x}^{3}z+y{z}^{3}+2\,{z}^{2}{x}^{2}+xy{z}^{2}-3\,x{y}^{2}z-z{x}^{2}y \right) \left( x-y \right) \left( x-z \right) +x \left( {y}^{3}+{z}^{3}+2\,x{y}^{2}-{x}^{2}y+{z}^{2}x-{y}^{2}z+2\,xyz \right) \left( y-z \right) ^{2}\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_1763 (Apache-2.0). Complete source proposition preserved; proof developed independently.