Eventual monotonicity conjecture for generalized partition ratios

Let $A\subset\mathbb{Z}_{>0}$, let $p^{(k)}(n)$ be the generalized partition functions associated with $A$, and define

$$\rho^{(k)}(n)=\frac{p^{(k+1)}(n)}{p^{(k)}(n)}=\frac{p^{(k)}(n)-p^{(k)}(n-1)}{p^{(k)}(n)}.$$

Assume that $A$ has properties $P_{k+1},P_{k+2},\dots$. Eventual monotonicity conjecture. The sequence $\rho^{(k)}(n)$ is eventually strictly decreasing.

This conjecture generalizes the eventual log-concavity of the classical partition function and would establish eventual monotonicity of the ratios for generalized partition functions under the stated property assumptions. The case $A=\mathbb{Z}_{>0}$ and $k=0$ follows from the eventual log-concavity theorem of DeSalvo and Pak, while the general case remains open in the supplied context.