PageRank fairness conjecture for homophilic and heterophilic two-colored PAMs

Consider the two-colored PAM with $m_1=m_2=m$, attachment kernel

$$\kappa(1,2)=\kappa(2,1)=\varepsilon,\qquad \kappa(1,1)=\kappa(2,2)=1-\varepsilon,$$

and minority-vertex probability $\mu_1$. PageRank fairness conjecture. For every sufficiently small $c>0$, there is some $\delta(c)>0$ such that, for every $m\in\mathbb{N}$, almost surely,

$$\limsup_{n\to\infty}\frac{R_\mathrm{mino}(G_n)}{n+n_0}<\mu_1$$

for every $0<\varepsilon<\frac12-\delta(c)$ and $\mu_1\in(0,\frac12)$, while

$$\liminf_{n\to\infty}\frac{R_\mathrm{mino}(G_n)}{n+n_0}>\mu_1$$

for every $\varepsilon>\frac12+\delta(c)$ and $\mu_1\in(0,\frac12)$. Thus, for sufficiently large graphs, minority PageRank is below the minority population proportion in the homophilic regime and above it in the heterophilic regime. This conjecture concerns whether PageRank systematically disadvantages or advantages the minority in the two-colored preferential attachment model; the supplied text gives no resolution beyond the stated claim.