Fixed-point and spinodal-curve conjecture for asymmetric community detection

Let $G:{\mathbb R}_+\to{\mathbb R}$ be the function

$$G(\mu)=\frac{\lambda}{(1-p)^2}{\mathbb E}\left[\frac{1}{p+(1-p)\exp(\sqrt{\mu}Z-\mu/2)}-1\right],$$

where $Z\sim\mathcal{N}(0,1)$, let $p^*=\frac{1}{2}-\frac{1}{2\sqrt{3}}$, and define the spinodal curve by

$$\lambda_{sp}(p)=\sup\{\lambda\geq0\mid 0\text{ is the unique fixed point of }G\}.$$

Fixed-point and spinodal-curve conjecture. (i) If $\lambda>1$, then $G$ has two fixed points, $0$ and $\alpha>0$; moreover, $0$ is unstable and $\alpha$ is stable. (ii) For $p^*\leq p\leq1/2$, $\lambda_{sp}(p)=1$. (iii) For $0\leq p<p^*$, $\lambda_{sp}(p)<1$, and if $\lambda_{sp}(p)<\lambda<1$, then $G$ has three fixed points $0<\beta<\alpha$; moreover, $0$ and $\alpha$ are stable and $\beta$ is unstable.

These properties are intended to establish that the spinodal curve is well defined and to describe the fixed-point structure governing reconstruction in the asymmetric stochastic block model. The source provides no resolution of this conjecture.