Expander-family conjecture for undirected prefix-reversal graphs

Let $\mathbb{P}_m(n)$ be the undirected prefix-reversal graph, which is $2n$-regular for $m>1$. A sequence of regular graphs is an expander family if there exists $\varepsilon>0$ such that every graph in the sequence has expansion ratio at least $\varepsilon$. Expander-family conjecture. Given $n\geq1$, the sequence of $2n$-regular graphs

$$\{\mathbb{P}_m(n)\}_{m>1}$$

indexed by $m$ is an expander family of graphs. The conjecture concerns expansion as $m$ varies with $n$ fixed; the source provides no resolution.