$\newcommand\si\sigma$This is not true. For instance, suppose that $\mu_1=\mu_2=\mu_3=0$ and
$$(\si_1,\si_2,\si_3)=\Big(\frac{1}{397},\frac{1}{161},\frac{5}{326}\Big).$$
Then the difference between the righ- and left-hand sides of your inequality is $-0.15709\ldots<0$.
Iosif Pinelis
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