Steady-state phase-difference conjecture for the two-community noisy Kuramoto model

Let $\theta$ denote the phase on the circle, let $\theta_{1}$ and $\theta_{2}$ be the average phases of the two communities, and write their phase difference as

$$\psi=\psi_{2}-\psi_{1}.$$

In the disorder-free two-community noisy Kuramoto model, assume $K_{1}=K_{2}=K$ and $L_{1}=L_{2}=L\neq 0$. Steady-state phase-difference conjecture. In the steady state, the phase difference can only be

$$\psi=0\quad\text{or}\quad\psi=\pi.$$

The claim is motivated by the expectation that the system maximizes the interaction strength between oscillators to achieve the highest synchronization in each community. The source provides heuristic rather than rigorous evidence.