Prove that $\sum\limits_{cyc}\frac{a^2}{a^3+2}\leq\frac{4}{3}$ if $abcd=1$

Let $a$, $b$, $c$ and $d$ be positive numbers such that $abcd=1$. Prove that: $$\frac{a^2}{a^3+2}+\frac{b^2}{b^3+2}+\frac{c^2}{c^3+2}+\frac{d^2}{d^3+2}\leq\frac{4}{3}$$ Vasc's LCF Theorem does not help here. Also I tried MV method, but without success.