Uniqueness of mean-field macroscopic state

Question

Suppose that the dynamics of a system obey $\dot{x_i}=-x_i+\sum_j w_{ij} \phi(x_j)$, where $w_{ij}$ are random but not iid. They can be for example correlated such that $<w_{ij} w_{ji}>\neq 0$. The fixed point solution of such a system is given by $x_i=\sum w_{ij} \phi(x_j)$. Now, we know that for a finite N, this system could have many different fixed point solutions, with different macroscopic states. I.e., the distribution of coordinates of different fixed point solutions could be different. Next, in the limit of $N \rightarrow \infty$, after we averaged over $w_{ij}$ we found the mean-field solution to be $x=z+\alpha \phi(x)$ where z is a random Gaussian variable and $\alpha$ is a function of the basic parameters of the problem. My question is whether the MF solution is the only possible macroscopic state. I.e., must all fixed point solutions have (up to a permutation in indices) the same coordinates distribution? Is it true to say that as N increases, the fixed-point solutions converge to the MF solution? Or is it just the average solution? Does the answer depend on the stability of the MF solution? Thank you