For positive $a$, $b$, $c$ with $abc=1$, show $\sum_{cyc} \left(\frac{a}{a^7+1}\right)^7\leq \sum_{cyc}\left(\frac{a}{a^{11}+1}\right)^7$

Question

I'm interested by the following problem :

Let $a,b,c$ be real positive numbers such that $abc=1$ and $a \leq 1$ and $b\leq 1$ then we have : $$\left(\frac{a}{a^7+1}\right)^7+\left(\frac{b}{b^7+1}\right)^7+\left(\frac{c}{c^7+1}\right)^7\leq \left(\frac{a}{a^{11}+1}\right)^7+\left(\frac{b}{b^{11}+1}\right)^7+\left(\frac{c}{c^{11}+1}\right)^7$$

First the equality occurs for $a=b=c=1$ .

Secondly It seems that it's not intuitive because the degree is higher on the RHS

Thirdly I have made the numerical test with Pari-Gp and I didn't found any counter-example .

If you have nice ideas it would be great.

Thanks in advance for your time .

Ps : I'm inspired by the following inequality For $abc=1$ prove that $\sum\limits_{cyc}\frac{a}{a^{11}+1}\leq\frac{3}{2}.$