 Solution

Neither Markov chain irreducibility nor fast convergence follows from detailed balance. On two states, the identity transition matrix satisfies detailed balance for the uniform probability distribution, but neither state can reach the other.

Even after imposing Markov chain irreducibility and aperiodicity, considerIt satisfies detailed balance for $\pi =(1/2,1/2)$. Its nontrivial eigenvalue is $1-2\epsilon$, and, from state zero, its total variation distance from $\pi$ after $n$ steps is $\frac{1}{2}(1-2\epsilon {)}^n$. This approaches zero arbitrarily slowly as $\epsilon$ decreases. Detailed balance without rapid convergence separates correctness of a stationary distribution from reachability and speed.