The relaxation time does not control the normal approximation of stationary additive functionals

Let X be a finite irreducible reversible Markov chain in continuous time with relaxation time τ₂, started from its stationary distribution, and let g be a function on its states with mean $0$ and variance $1$ under that distribution. Aldous and Fill asked whether the Kolmogorov distance between ∫₀ᵗg(Xᵤ)\,du/(σ t1/2), where σ² is the asymptotic variance rate, and the standard normal law is bounded by a function of max|g| and t/τ₂ alone that tends to $0$ as t/τ₂→∞. It is not. For every b≥1 and s≥2b there are a chain on four states with τ₂=1 and a function g of mean $0$ and variance $1$ with max|g|=b such that ∫₀ˢg(Xᵤ)\,du≤0 with probability at least $1-4/s$, so at $t=s$ the distance is at least $1/2-4/s$ whatever positive constant the integral is divided by.