A reciprocal-total product inequality with a pair-product correction

$a,b,c,d>0$ ,prove

$\left( a+b+c+d \right) \left( {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}+{\frac {1}{d}} \right) +8\geq 9\,{\frac { \left( a+b+c+d \right) ^{2}}{ab+bc+cd+ad+ac+bd}}$

Source: InternLM Lean-Workbook, record lean_workbook_9999 (Apache-2.0). Complete source proposition preserved; proof developed independently.