The $f(6)$ feedback vertex set conjecture

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. The $f(6)$ conjecture.

$$f(6)=\frac{4}{7}.$$

The paper has already established the corresponding exact values for $f(4)$ and $f(5)$, while the value at $k=6$ is proposed as the next case and remains open in the supplied text.