Let $\mathrm{X}\sim\mathcal{N}_{3}(\boldsymbol{\mu},\mathrm{\Sigma})$ where
\begin{equation} \boldsymbol{\mu} = n[(\mu_1-\mu_2)\sqrt{\xi_1\xi_2/(\xi_1+\xi_2)}, (\mu_1-\mu_3)\sqrt{\xi_1\xi_3/(\xi_1+\xi_3)},(\mu_2-\mu_3)\sqrt{\xi_2\xi_3/(\xi_2+\xi_3)}] \end{equation} and \begin{equation} \mathrm{\Sigma}= \begin{bmatrix}1 & \sqrt{\frac{\xi_2\xi_3}{(\xi_1+\xi_2)(\xi_1+\xi_3)}} & -\sqrt{\frac{\xi_1\xi_3}{(\xi_1+\xi_2)(\xi_2+\xi_3)}} \\ \sqrt{\frac{\xi_2\xi_3}{(\xi_1+\xi_2)(\xi_1+\xi_3)}} & 1 & \sqrt{\frac{\xi_2\xi_1}{(\xi_1+\xi_3)(\xi_2+\xi_3)}}\\ -\sqrt{\frac{\xi_1\xi_3}{(\xi_1+\xi_2)(\xi_2+\xi_3)}} & \sqrt{\frac{\xi_2\xi_1}{(\xi_1+\xi_3)(\xi_2+\xi_3)}} & 1 \end{bmatrix} \end{equation} $\xi_i\geq 0$ for $i=1,2,3$ and $\xi_1+\xi_2+\xi_3 = 1$ and $n$ is some positive integer.
Consider the following function \begin{equation} \phi(\boldsymbol{\mu},\mathrm{\Sigma}) = \int_{-q}^{q}\int_{-q}^{q}\int_{-q}^{q}f_{\mathrm{X}}(\boldsymbol{\mu},\mathrm{\Sigma})d\boldsymbol{x} \end{equation} where $f_{\mathrm{X}}$ denotes the density of multivariate normal random variable and $q>0$. Let us assume that $|\mu_1-\mu_2| = \max|\mu_i-\mu_j|$ for $i\neq j$.
Numerical results suggest that the minimum value of $\phi(\boldsymbol{\mu},\mathrm{\Sigma})$ for given $\boldsymbol{\mu}$, $n$ and $q$ with respect to $\boldsymbol{\xi}$ is attained for $(\xi_1,\xi_2,\xi_3) = (1/2,1/2,0)$. Is there any theoretical proof of this claim? I tried stochastic ordering approach but failed.
