Global phase in stochastic Kuramoto model with vanishing natural frequencies

Question

This is a question that popped up when I was looking at the noisy Kuramoto model with vanishing natural frequencies. Specifically, the model I am considering is the following \begin{equation} \mathrm{d}\theta_i=\frac{K}{N}\sum_{j=1}^N\sin(\theta_j-\theta_i)\mathrm{d}t+\sqrt{2D}\mathrm{d}W_i \end{equation} where $W_i$ are standard Brownian motions. Clearly, the mean angle $\bar{\theta}=N^{-1}\sum_j\theta_j$ is just a Brownian motion, and the steady-state distribution of $\bar{\theta}$ is thus simply the uniform distribution. However, does this imply that $S=N^{-1}\sum_{j=1}^Ne^{i\theta_j}$ has vanishing expectation with respect to the steady-state measure? I.e. is \begin{equation} \mathbb{E}e^{i\theta_j}=0? \end{equation} Of course, one could still have synchronization, e.g. with a global phase $\arg S\sim U(-\pi,\pi)$. This probably has an answer somewhere, I just can't seem to find it.