Phase-alignment conjecture for the three-user semi-symmetric Gaussian interference channel

Consider the three-user semi-symmetric complex Gaussian interference channel

$$Y_1=X_1+g_1X_2+g_2X_3+Z_1,$$ $$Y_2=X_2+g_1X_3+g_2X_1+Z_2,$$ $$Y_3=X_3+g_1X_1+g_2X_2+Z_3,$$

where $P_k=P$ for $k=1,2,3$, and write $\phi(g_1)$ and $\phi(g_2)$ for the phases of $g_1$ and $g_2$. Phase-alignment conjecture. The sum capacity may take its maximum values along the intersection of the following two types of lines:

$$2\phi(g_1)-\phi(g_2)+\pi=0 \mod 2\pi,$$ $$2\phi(g_2)-\phi(g_1)+\pi=0 \mod 2\pi.$$

It can be minimized along any of the following two types of lines:

$$2\phi(g_1)-\phi(g_2)=0 \mod 2\pi,$$ $$2\phi(g_2)-\phi(g_1)=0 \mod 2\pi.$$

The claim is based on numerical observations of the upper bound in several settings; because the observed behavior is not matched by a lower bound, the phase dependence may not be fundamental and the conjecture remains open.