Hong–Zhang's log-concavity conjecture for power coefficients of the divisor-sum series

Let

$$f(q)=\sum_{n=1}^{\infty}\sigma(n)\,\frac{q^n}{n}$$

and, for $m\in\mathbb{N}$, let $b_{m,n}$ denote the coefficient of $q^n$ in $f(q)^m$, with $b_{m,n}=0$ otherwise. Hong–Zhang's conjecture. There exists a constant $C>1$ such that, for all $m\geq 2$ and $1\leq n\leq C^m$,

$$b_{m,n}^2\geq b_{m,n-1}b_{m,n+1}.$$

This asserts log-concavity in an initial range of the coefficient sequence for every power of the divisor-sum generating series. The conjecture is attributed to Hong and Zhang; the material supplied here gives no resolution, so its status remains open.