A quartic product inequality under three pairwise constraints

If $x, y, z > 0$ are real numbers and $z(x+y+z) \ge xy$ and $x(x+y+z) \ge yz$ and $y(x+y+z) \ge zx$ , prove that:

$(x^2+y^2+z^2)(x+y+z)^2 \ge 8xyz(x+y+z)+x^2y^2+y^2z^2+z^2x^2 \ \ ;$

Greetings from Lorian Saceanu!

12-March 2017

Source: InternLM Lean-Workbook, record lean_workbook_plus_69795 (Apache-2.0). Complete source proposition preserved; proof developed independently.