Chern–Fu–Tang conjecture on log-concavity of k-colored partitions

Let $p_{-k}(n)$ denote the number of $k$-colored partitions of $n$, with $p_{-k}(n)=P_n(k)$, where $P_0(x)=1$ and

$$P_n(x):=\frac{x}{n}\sum_{j=1}^n\sigma(j)P_{n-j}(x),\qquad \sigma(j):=\sum_{d\mid j}d.$$

Chern–Fu–Tang conjecture. Let $n>m\geq 1$ and $k\geq 2$. Except for $(k,n,m)=(2,6,4)$,

$$p_{-k}(n-1)p_{-k}(m+1)\geq p_{-k}(n)p_{-k}(m).$$

This is a Bessenrodt–Ono-type log-concavity inequality for colored partition numbers. The source does not provide a resolution status for the conjecture.