Asymptotic feedback vertex set conjecture for bounded-degree digraphs

For any integer $k\ge 2$, let $f(k)$ be the supremum of ${\rm fvs}(D)/|V(D)|$ over all orgraphs $D$ with maximum degree at most $k$, where ${\rm fvs}(D)$ denotes the minimum feedback vertex set size. Asymptotic bounded-degree conjecture. There exists a constant $\theta>0$ such that for all $k\ge 3$,

$$f(k)\le 1-\theta\cdot\frac{\log_2 k}{k}.$$

The paper proves exact values for $f(4)$ and $f(5)$ and notes that the conjectured upper bound would be best possible up to the constant, because tournaments provide matching logarithmic-order lower bounds. The conjecture is presented as an open problem for larger maximum degrees.