Finite-difference sign conjecture for the broken $k$-diamond partition function

Let $\Delta_k(n)$ be the number of the $k$-diamond broken partition of $n$, and let $D$ denote the difference operator with respect to $n$.

Finite-difference sign conjecture. For any $k\geq 1$ and $r\geq 1$, there exists a positive integer $n_k(r)$ such that, for $n\geq n_k(r)$,

$$(-1)^{r-1}D^r\log \Delta_k(n)>0.$$

This extends the proved cases for $k=1,2$ and third differences, and is analogous to the positivity of finite differences of the logarithm of the partition function. The general assertion remains open.