Occupation sums of the harmonic descent chain and slowly decaying weights

Let $a(n,i)$ be the probability that the harmonic descent chain started at $n$ visits $i$, and $a(i)=\lim_n a(n,i)$. Aldous and Janson asked whether $\sum_{i=2}^n a(n,i)f(i)\to\sum_{i\ge2}a(i)f(i)$ for every non-negative $f$ of polynomial growth with a finite limit sum, and Brandenberger, Chin and Mossel proved this when $f(i)=O(i^{-\varepsilon})$. We show that for every $\delta>0$ there is an $f$ with $f(i)=O((\log i)^{-1+\delta})$ and $\sum_i a(i)f(i)<\infty$ for which the occupation sums are unbounded, along values of $n$ with $f(n)=0$. The cause is that the chain started at $n$ makes, on average, at least of order $\log n/\log\log n$ visits to a window of width $n/(\log n)^2$ just below $n$.