A parametric cyclic quartic inequality

Prove the following inequality for real numbers $a,b,c$ :

$a^4+b^4+c^4+r(a^2b^2+b^2c^2+c^2a^2)+(p+q-r-1)(a^2bc+ab^2c+abc^2)\ge p(a^3b+b^3c+c^3a)+q(ab^3+bc^3+ca^3)$

Where $p,q,r$ are real numbers which satisfy $3(1+r)\ge p^2+pq+q^2$

$\sum_{cyc}(a^4+ra^2b^2+(p+q-r-1)a^2bc-pa^3b-qa^3c)\geq0\Leftrightarrow$

$\Leftrightarrow\sum_{cyc}(2a^4+2ra^2b^2+2(p+q-r-1)a^2bc-(p+q)(a^3b+a^3c))\geq(p-q)\sum_{cyc}(a^3b-a^3c)$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_46959 (Apache-2.0). Complete source proposition preserved; proof developed independently.