Given $a,b,c>0$, prove that $$(a+\frac{bc}{a})(b+\frac{ca}{b})(c+\frac{ab}{c})\geq 4 \sqrt[3]{(a^3+b^3)(b^3+c^3)(c^3+a^3)}.$$
I noticed that by AM-GM the LHS $\leq 4[\frac{2(a^3+b^3+c^3)}{3}]^3 =\frac{8}{3}(a^3+b^3+c^3)$ and the RHS $=2abc+\frac{a^2b^2}{c}+\frac{b^2c^2}{a}+\frac{c^2a^2}{b}+a^3+b^3+c^3.$ Therefore proving $2abc+\frac{a^2b^2}{c}+\frac{b^2c^2}{a}+\frac{c^2a^2}{b}\geq \frac{5}{3}(a^3+b^3+c^3)$ would suffice. However, I've found a counterexample to that last inequality. So how could I've done better?
Because by AM-GM $$\prod_{cyc}\left(a+\frac{bc}{a}\right)=\frac{1}{abc}\sqrt{\prod_{cyc}(a^2+bc)(b^2+ac)}=$$ $$=\frac{1}{abc}\sqrt{\prod_{cyc}(c(a^3+b^3)+ab(c^2+ab))}\geq\frac{1}{abc}\sqrt{\prod_{cyc}2\sqrt{c(a^3+b^3)(c^2+ab)}}.$$ Can you end it now?