Half-induced-forest conjecture for graphs on surfaces

For a graph $G$ with $n$ vertices and genus $g$, let $\alpha_1(G)$ denote the order of a largest induced $1$-degenerate subgraph of $G$, equivalently a largest induced forest. Half-induced-forest conjecture. For every non-negative integer $g$, there exists a constant $f(g)$ such that every graph $G$ of genus $g$ satisfies

$$\alpha_1(G)\geq \frac{n}{2}-f(g).$$

The conjecture is known for triangle-free planar graphs, where the source cites the stronger bound $\alpha_1(G)\geq (6n+7)/11$. It remains open for graphs on surfaces in general.