Prove or disprove $(a^2+b^2)(b^2+c^2)(c^2+a^2)\leq(a+b)(b+c)(c+a)$

Question

Prove or disprove:

If $a, b, c \in (0, \infty)$ such that $a^2+b^2+c^2=3$ then $$(a^2+b^2)(b^2+c^2)(c^2+a^2)\leq(a+b)(b+c)(c+a).$$

All my attempts to prove inequality have been unsuccessful. Maybe someone has an idea. Thank you very much!

Answer 1

It's wrong! Try $c\rightarrow0^+$ and $a=b\rightarrow\sqrt{1.5}$