For $a,b,c$ positive real numbers can it be true that: $(ab+bc+ca)^3 \ge (a^2+2b^2)(b^2+2c^2)(c^2+2a^2)$ For $a,b,c$ positive real numbers can it be true that: $$(ab+bc+ca)^3 \ge (a^2+2b^2)(b^2+2c^2)(c^2+2a^2)$$
It really seems unlikely because it reminds me the rearrangement inequality when the direction is flipped. 

$\prod\limits_{cyc}(a^2+2b^2)\geq(ab+ac+bc)^3\Leftrightarrow\sum\limits_{cyc}(2a^4b^2+4a^4c^2-a^3b^3-3a^3b^2c-3a^3c^2b+a^2b^2c^2)\geq0$, 
which is true by Schur and AM-GM:
$\sum\limits_{cyc}(2a^4b^2+4a^4c^2-a^3b^3-3a^3b^2c-3a^3c^2b+a^2b^2c^2)\geq$
$\geq\sum\limits_{cyc}(2a^4c^2+3a^3b^3-3a^3b^2c-3a^3c^2b+a^2b^2c^2)=$
$=3\sum\limits_{cyc}(a^3b^3-a^3b^2c-a^3c^2b+a^2b^2c^2)+2\sum\limits_{cyc}(a^4c^2-a^2b^2c^2)\geq0$.
