The sub-nine acyclic list-colouring conjecture for locally planar graphs

Let $G$ be a graph embedded in a surface $\Sigma$, and let $g(\Sigma)$ denote the genus of $\Sigma$. A graph is $\rho$-locally planar if the relevant local planarity condition holds with parameter $\rho$. An acyclic $k$-list-colouring is a list-colouring from lists of size $k$ in which every cycle receives at least three colours. Sub-nine acyclic list-colouring conjecture. There exists an integer $k<9$ and a value $\rho=\rho(g(\Sigma))$ such that every $\rho$-locally planar graph embedded in $\Sigma$ is acyclically $k$-list-colourable.

The paper proves the corresponding bound with $9$ colours and notes that the list size is probably not optimal. It suggests that $k=8$ may be attainable, while $k=7$ is also considered possible; the optimal value remains unclear.