The largest Kemeny constant and the largest relaxation time of a graph

For the simple random walk on a connected graph with $n$ vertices, let $\tau_0$ be Kemeny's constant and $\tau_2$ the relaxation time. Aldous and Fill conjectured that the maxima of both quantities over all connected $n$-vertex graphs are asymptotic to $n^3/54$. We prove this. Every connected graph on $n$ vertices satisfies $\tau_2\le\tau_0\le n^3/54+\tfrac{37}{8}n^{5/2}$, and the barbell made of two cliques on about $n/3$ vertices joined by a path on about $n/3$ vertices has $\tau_2\ge(n-2)(n-5)(n+3)/54$. The upper bound comes from the electrical formula for Kemeny's constant: the vertices near high-degree vertices are grouped into components of small total diameter, the rest of the graph is compared with a tree, and each edge of the tree is shown to contribute at most $s^2/8$ plus a lower-order term, where $s$ is the number of grouped vertices.