Aldous–Fill conjecture on maximum relaxation time

Let $G$ be a connected regular graph on $n$ vertices, and let $\tau$ denote the relaxation time of its random walk, defined by

$$\tau=\frac{1}{1-\eta_2},$$

where $\eta_2$ is the second-largest eigenvalue of the transition matrix. Aldous–Fill conjecture. Over all connected regular graphs on $n$ vertices,

$$\max \tau=(1+o(1))\frac{3n^2}{2\pi^2}.$$

The conjecture concerns the slowest-mixing random walk among connected regular graphs and is equivalent, within each degree, to minimizing the spectral gap. The paper proves the corresponding asymptotic result for quartic graphs, so the conjecture follows for degree $4$, while the general statement is not resolved here.