Improved upper bound for feedback vertex sets in graphs of even degeneracy

Let $k$ be an even degeneracy bound, and let $G$ be an $n$-vertex graph of degeneracy $k$. Write $f$ for the size of a minimum feedback vertex set of $G$.

There is an $\varepsilon>0$ such that

$$f\leq \left(\frac{k}{k+2}-\varepsilon\right)n.$$

The paper gives an easy upper bound with coefficient $\frac{k}{k+2}$ and constructs non-trivial lower bounds, but does not essentially improve this upper bound; the conjecture asserts that a uniform improvement is possible.