Equicorrelated normals: sums of two coordinates share one law

Let $X=(X_1,X_2,X_3,X_4)\sim N_4(\underline0,\Sigma)$ where

$$\Sigma=\begin{bmatrix}1&\rho&\rho&\rho\ \rho&1&\rho&\rho\ \rho&\rho&1&\rho\ \rho&\rho&\rho&1\end{bmatrix}$$

is positive definite. Then which of the following statements is true?