Forest-cut conjecture for sparse graphs

Let $G$ be a finite, simple, undirected graph of order $n$, and call a vertex set a forest cut if it is a vertex cut whose induced subgraph is a forest. Forest-cut conjecture. If $G$ has less than $3n-6$ edges, then $G$ has a forest cut. Maximal planar graphs do not have such vertex cuts, so the density condition would be best possible; the conjecture is verified for planar graphs, and the paper proves the weaker bound that less than $\frac{11}{5}n-\frac{18}{5}$ edges suffice in general.