The linear color function equals the DP color function for prime powers

Let $k$ be a power of a prime. For a graph $G$, let $P_{\mathcal{L}_k}(G,k)$ be the minimum number of colorings over all $\mathcal{L}_k$-labelings, where $\mathcal{L}_k$ is the collection of $\mathbb{F}_k$-linear permutations of $\mathbb{F}_k$, and let $P_{DP}(G,k)$ denote the DP color function. The linear-color-function conjecture. For every graph $G$,

$$P_{\mathcal{L}_k}(G,k)=P_{DP}(G,k).$$

The conjecture would identify the restricted linear-labeling model with the general DP-coloring model for prime-power numbers of colors. It is motivated by the fact that the authors know of no graph for which the lower bound obtained for $P_{\mathcal{L}_k}(G,k)$ fails for the DP color function; the source states that it is open for prime powers $k\geq 4$ and true for $k=2,3$.