This is a followup to my question Cover times and hitting times of random walks.
Consider a random walk on an undirected graph with $n$ vertices which, at each step, moves to a uniformly random neighbor. Define $T(u,v)$ to be the expected time until such a walk, starting from $u$, arrives at $v$, and let $T = \max_{u,v} T(u,v)$. Define $G(u)$ to be the expected time until such a walk, starting from $u$, visits every vertex and let $G = \max_u G(u)$. Is it true that $G \leq cT \log^k n$ for some constants $c,k$ which do not depend on the graph or on $n$?