It take me a little bit of time to create this :
Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$\sqrt{\frac{(ab)^2+(bc)^2+(ca)^2}{a^4+b^4+c^4}}\Big(\sum_{cyc}\frac{a^2}{11a+b}\Big)+\frac{(a-b)^2}{12}\leq \frac{1}{12}$$
First of all we have two equality case when $a=b=c=\frac{1}{3}$ and $a=1$
I have tried to delete the square root and after a full expanding to achieve with the Buffalo's ways.But I cannot conclude since we have negatives terms .I have not try Lagrange multiplier because I'm not familiar with this method but I think that the system resulting is awful .
It's wrong.
Try $$(a,b,c)=\left(\frac{1}{2401},\frac{1400}{2401},\frac{1000}{2401}\right).$$ In this case we have: $$\frac{1}{12}-LHS=-0.000718...$$