The given inequality is equivalent to, $$2ab(a-b)^2+2bc(b-c)^2+2ca(c-a)^2 \enspace \geq \enspace (ab+bc+ca)^2$$ $$\iff \quad \quad 2ab(a-b)^2 \enspace+\enspace 2bc(b-c)^2\enspace+\enspace 2ca(c-a)^2 \enspace \geq \enspace a^2b^2\enspace+\enspace b^2c^2\enspace+\enspace c^2a^2\enspace+\enspace 2a^2bc\enspace+\enspace 2ab^2c\enspace+\enspace 2abc^2$$ $$\iff \quad\quad 2\sum_{\text{cyc}} ab(a-b)^2 \enspace \geq \enspace \sum_{\text{cyc}} a^2b^2 \enspace + \enspace 2\sum_{\text{cyc}}a^2bc $$ $$\iff \quad \quad 2\sum_{\text{cyc}} a^3b \enspace-\enspace 4 \sum_{\text{cyc}} a^2b^2 \enspace+\enspace2\sum_{\text{cyc}} a^3c \enspace \geq \enspace \sum_{\text{cyc}} a^2b^2 \enspace + \enspace 2\sum_{\text{cyc}}a^2bc $$ $$\iff \quad \quad 2\sum_{\text{cyc}} a^3b \enspace+\enspace 2\sum_{\text{cyc}} a^3c \enspace \geq \enspace5 \sum_{\text{cyc}} a^2b^2 \enspace + \enspace 2\sum_{\text{cyc}}a^2bc $$ Using the condition $2a^3b+2b^3c+2c^3a=a^2b^2+b^2c^2+c^2a^2$, this is equivalent to $$\iff \quad \quad \left(2\sum_{\text{cyc}} a^3b \enspace+\enspace 2\sum_{\text{cyc}} a^3c\right) \enspace + \enspace \left(6\sum_{\text{cyc}} a^3b \right) \enspace \geq \enspace \left(5\sum_{\text{cyc}} a^2b^2 \enspace + \enspace 2\sum_{\text{cyc}}a^2bc\right)\enspace + \enspace \left( 3\sum_{\text{cyc}} a^2b^2 \right) $$ $$\iff \quad \quad 8\sum_{\text{cyc}} a^3b \enspace+\enspace 2\sum_{\text{cyc}} a^3c \enspace \geq \enspace 8\sum_{\text{cyc}} a^2b^2 \enspace + \enspace 2\sum_{\text{cyc}}a^2bc$$ $$\iff \quad \quad 4\sum_{\text{cyc}} a^3b \enspace+\enspace \sum_{\text{cyc}} a^3c \enspace \geq \enspace 4\sum_{\text{cyc}} a^2b^2 \enspace + \enspace \sum_{\text{cyc}}a^2bc$$ $$\iff \quad \quad 4\sum_{\text{cyc}} a^3b \enspace+\enspace \sum_{\text{cyc}} a^3c \enspace \geq \enspace 4\sum_{\text{cyc}} a^2b^2 \enspace + \enspace \sum_{\text{cyc}}a^2bc$$ $$\iff \quad \quad 4\sum_{\text{cyc}} a^3b \enspace+\enspace \sum_{\text{cyc}} ab^3+ \sum_{\text{cyc}} abc^2- 4 \sum_{\text{cyc}} a^2b^2 - 4\sum_{\text{cyc}} a^2bc + 2\sum_{\text{cyc}} ab^2c \enspace \geq \enspace 0$$ $$ \iff \quad \quad \sum_{\text{cyc}} ab(2a-b-c)^2 \enspace \geq \enspace 9$$ which is obviously true.