The logarithmic edge-width conjecture for acyclic list-colouring locally planar graphs

Let $\Sigma$ be a surface and let $g(\Sigma)$ denote its genus. Fix an integer $k$ for which there is a value $\rho_0=\rho_0(g(\Sigma))$ such that every $\rho_0$-locally planar graph embedded in $\Sigma$ is acyclically $k$-list-colourable. Logarithmic edge-width conjecture. There exists

$$\rho=O(\log g(\Sigma))$$

such that every $\rho$-locally planar graph embedded in $\Sigma$ is acyclically $k$-list-colourable.

The paper's theorem improves the relevant edge-width dependence from exponential to linear in the genus, and this conjecture proposes a further logarithmic bound while preserving any fixed admissible list size $k$. The supplied text does not indicate that the conjecture has been resolved.