List-colouring bound for graphs with sparse neighbourhoods

Let $G$ be a graph of maximum degree $\Delta$ such that the neighbourhood of every vertex spans at most $\Delta^2/f$ edges. List-colouring conjecture. For every $\varepsilon>0$, there exists $f_0$ such that, whenever $f_0\le f\le \Delta^2+1$, the list chromatic number of $G$ is at most

$$(2+\varepsilon)\Delta/\log f.$$

The paper presents this as a strengthening suggested by its fractional-colouring theorem and related results; the supplied source does not establish the list-colouring bound, and its current resolution status is unclear.