Hard inequality :$\Big(\frac{1}{a^2+b^2}\Big)^2+\Big(\frac{1}{b^2+c^2}\Big)^2+\Big(\frac{1}{c^2+a^2}\Big)^2\geq \frac{3}{4}$ I have a hard problem this is it :

Let $a,b,c>0$ such that $a^ab^bc^c=1$ then we have :
  $$\Big(\frac{1}{a^2+b^2}\Big)^2+\Big(\frac{1}{b^2+c^2}\Big)^2+\Big(\frac{1}{c^2+a^2}\Big)^2\geq \frac{3}{4}$$

I try to use Jensen's ienquality applied to the function $f(x)=\frac{1}{x^2}$ but it doesn't works for all the values .
if we apply Am-Gm it's like Jensen's inequality so I forget this ways .
Maybe If we apply Karamata's inequality but I didn't found the right majorization .
I try also to use Muirhead inequality but without success .
So I'm a bit lost with that if you have a hint or a full answer I will be happy to read your work . 

Hint.
Making $b = \lambda a, c = \mu a$ we have
$$
\frac{1}{(1+\lambda^2)^2}+\frac{1}{(\lambda^2+\mu^2)^2}+\frac{1}{(\mu^2+1)^2}\ge \frac 34 a^4
$$
and
$$
a = \frac{1}{(\mu^{\mu}\lambda^{\lambda})^{\frac{1}{1+\lambda+\mu}}}
$$
then if 
$$
f(\lambda,\mu) = \frac{1}{(1+\lambda^2)^2}+\frac{1}{(\lambda^2+\mu^2)^2}+\frac{1}{(\mu^2+1)^2}- \frac 34 \frac{1}{(\mu^{\mu}\lambda^{\lambda})^{\frac{4}{1+\lambda+\mu}}}
$$
now we need to demonstrate that 
$$
\min f_{\lambda,\mu} \ge 0
$$
Note that 
$$
f(\lambda,\lambda) \le f(\lambda,\mu)
$$
It is easy to verify also that along $\lambda = \mu$
$$
f(1,1) = 0\\
f'(1,1) = 0\\
f''(1,1) > 0
$$
