The unique-fixed-point conjecture for local belief propagation

Let $\rho$ denote the cluster-size parameter, let $\theta$ and $u$ be model parameters, let $\theta+ u$ determine the recursion parameter $u$, and let $\theta+ u h(v)$ be the scalar belief-propagation recursion, where $h$ is the function defined in the paper. A fixed point is a value $v$ satisfying

$$v=\theta+\lambda h(v).$$

Unique-fixed-point conjecture. If $mu=nu$, then this recursion has a unique fixed point for every

$$\rho\in(0,1/2)\cup(1/2,1).$$

A unique fixed point would imply that local belief propagation is optimal in the corresponding unequal-cluster-size cases. The claim is motivated by numerical experiments and is presented in the paper as an open problem; its resolution is not supplied here.