The variation threshold, Kemeny's constant and stationary times of reversible Markov chains

For finite reversible Markov chains, Aldous and Fill asked whether the variation threshold τ₁ is at most the average hitting time τ₀ (Kemeny's constant), and whether the relaxation time τ₂ is at most τ₁⁽²⁾, the largest over initial states of the least mean of a stopping time at which the chain has its stationary distribution. The answer to the first question is no: a birth and death chain on three states with a strong drift has τ₁>τ₀, and on the analogous chains with $k+1$ states the ratio τ₁/τ₀ tends to tₖ/k as the drift grows, where tₖ is the point at which the survival function of the Gamma$(k,1)$ distribution equals e⁻¹; for $k=4$ this limit is 1.0879…. The answer to the second question is yes, in continuous and in discrete time.