Equitable list arboricity conjecture for powers of paths

Let $n,p\in\mathbb{N}$ with $n\geq p+1$, and let $P_n^p$ denote the $p$th power of the path $P_n$; that is, vertices at distance at most $p$ in $P_n$ are adjacent in $P_n^p$. Set $G=P_n^p$.

Path-power equitable list arboricity conjecture. The graph $G$ is equitably $k$-list arborable if and only if

$$k\geq \left\lceil\frac{p+1}{2}\right\rceil.$$

The lower bound follows from the complete subgraph on $p+1$ vertices, while the supplied text says that the conjecture remains open for powers of cycles, not that this path-power conjecture itself has been resolved.