Existence of a universal amplifier of selection

Moran Birth-death processes are stochastic processes defined as follows. Consider a connected graph $G$ and a parameter $r \in \mathbb{R}$ strictly greater than $1$. We start with a partition of the vertices of $G$ into two types: mutant vertices, that have fitness $r$, and resident vertices, that have fitness $1$. At each step, a random vertex is chosen with probability proportional to its fitness, and spreads its type to an adjacent vertex chosen uniformly at random. With probability $1$, the process eventually reaches either fixation of the mutation (all the vertices are mutant) or extinction (all the vertices are resident). The fixation probability of a vertex $v$ of $G$ is the probability that the process starting with a single mutant at $v$ eventually reaches fixation. The fixation probability of each vertex of the complete graph on $n\in \mathbb{N}$ vertices is

$$p_n = \frac{1-r^{-1}}{1-r^{-n}}.$$

Open problem: Does there exist a connected graph $G$ with $n \in \mathbb{N}$ vertices such that the fixation probability of each vertex of $G$ is strictly greater than $p_n$?