$\newcommand\si\sigma$This is not true. For instance, suppose that $\mu_1=\mu_2=\mu_3=0$ and 
$(\si_1,\si_2,\si_3)=(1,2,6).$  
Then $d(P, Q) + d(Q, R) - d(P, R)$ is $-263/1764<0$.