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$\newcommand\si\sigma$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$$d(P, Q) + d(Q, R) - d(P, R)=-263/1764<0$.
$\newcommand\si\sigma$This is not true. For instance, suppose that $\mu_1=\mu_2=\mu_3=0$ and
$(\si_1,\si_2,\si_3)=(1,2,6).$
Then $d(P, Q) + d(Q, R) - d(P, R)$ is $-263/1764<0$.
$\newcommand\si\sigma$This is not true. For instance, suppose that $\mu_1=\mu_2=\mu_3=0$ and
$(\si_1,\si_2,\si_3)=(1,2,6).$
Then $d(P, Q) + d(Q, R) - d(P, R)=-263/1764<0$.
$\newcommand\si\sigma$This is not true. For instance, suppose that $\mu_1=\mu_2=\mu_3=0$ and
$(\si_1,\si_2,\si_3)=(1,2,6).$
Then the difference between the right- and left-hand sides of your inequality$d(P, Q) + d(Q, R) - d(P, R)$ is $-263/1764<0$.
$\newcommand\si\sigma$This is not true. For instance, suppose that $\mu_1=\mu_2=\mu_3=0$ and
$(\si_1,\si_2,\si_3)=(1,2,6).$
Then the difference between the right- and left-hand sides of your inequality is $-263/1764<0$.
$\newcommand\si\sigma$This is not true. For instance, suppose that $\mu_1=\mu_2=\mu_3=0$ and
$(\si_1,\si_2,\si_3)=(1,2,6).$
Then $d(P, Q) + d(Q, R) - d(P, R)$ is $-263/1764<0$.
$\newcommand\si\sigma$This is not true. For instance, suppose that $\mu_1=\mu_2=\mu_3=0$ and
$$(\si_1,\si_2,\si_3)=\Big(\frac{1}{397},\frac{1}{161},\frac{5}{326}\Big).$$$(\si_1,\si_2,\si_3)=(1,2,6).$
Then the difference between the righright- and left-hand sides of your inequality is $-0.15709\ldots<0$$-263/1764<0$.
$\newcommand\si\sigma$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 $-0.15709\ldots<0$.
$\newcommand\si\sigma$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 the difference between the right- and left-hand sides of your inequality is $-263/1764<0$.
