A conjecture about power sum : $e^{ab}+e^{bc}+e^{ca}\geq 3e^{\sqrt{abc}}$ and $a+b+c=3$

By Jensen $$e^{ab}+e^{ac}+e^{bc}\geq3e^{\frac{ab+ac+bc}{3}}\geq3e^{\sqrt{abc}}$$ because the last inequality it's $$ab+ac+bc\geq\sqrt{3(a+b+c)abc}$$ or after squaring of the both sides $$\sum_{cyc}c^2(a-b)^2\geq0.$$ The second inequality is wrong.

Try $n=4$, $a_1=a_3=\frac{1}{4}$ and $a_2=a_4=\frac{7}{4}.$