The inequality is equivalent to: $2\sum x^2+ \sum x^2y^2+\sum_{cyc} (\frac{y}{x}-2)^2 \geq 12 <=> (x+y+z)^2+\sum x^2 + \sum(xy-1)^2 + \sum_{cyc} (\frac{y}{x}-2)^2 \geq 15 <=> 9+\sum x^2 + \sum(xy-1)^2 + \sum_{cyc} (\frac{y}{x}-2)^2 \geq 15 <=> \sum x^2 + \sum(xy-1)^2 +\sum_{cyc} (\frac {y}{x}-2)^2 \geq 6$. We have that $\sum x^2 + \sum(xy-1)^2 + \sum_{cyc} (\frac {y}{x}-2)^2 \geq \sum x^2 + \sum_{cyc} (\frac {y}{x}-2)^2 \geq 2\sum_{cyc} |x(\frac {y}{x}-2)| \geq 2 |\sum_{cyc} x(\frac {y}{x}-2)|=6$, as needed.