Dual-DP Shameful inequality conjecture

Let $G$ be a graph on $n$ vertices, and let $P^{*}_{DP}(G,k)$ denote its dual DP color function, the maximum number of colorings over all full $k$-fold covers of $G$. Dual-DP Shameful inequality conjecture. For every $n$-vertex graph $G$ and every $k\in\mathbb{N}$ satisfying $k\geq n-1$,

$$\frac{P^{*}_{DP}(G,k+1)}{(k+1)^n}\geq\frac{P^{*}_{DP}(G,k)}{k^n}.$$

The text motivates this as an expected extension of the Shameful inequality: it notes counterexamples for smaller values of $k$, while giving no resolution of the stated range, so the conjecture is open.