Induced forest bound for graphs of prescribed girth

Let $G$ be a graph on $n$ vertices with $m$ edges, and let $\alpha_1(G)$ denote the order of a largest induced $1$-degenerate subgraph of $G$, equivalently a largest induced forest. Induced forest girth conjecture. For every integer $k\geq 3$, if $G$ has girth at least $k$, then

$$\alpha_1(G)\geq n-\frac{m}{k}.$$

The source notes that the bound would be best possible, as witnessed by a disjoint union of cycles of length $k$, and that the case of girth $3$ is trivial. The general assertion is presented as open.