D=1 simple-graph degree-restricted forest partition

Prove that every simple graph that is \((9/5,2/5)\)-sparse admits a \((1)\)-coloring, namely a vertex partition into a forest of maximum degree at most 1 and a forest.

Background

The paper establishes an exact sparsity bound for the degree-restricted two-forest partition problem on simple graphs for every integer $D\geq 2$. The authors explicitly identify the unresolved case $D=1$, corresponding to partitioning the vertices into a matching and a forest.

They state that their methods do not prove the conjectured $(9/5,2/5)$-sparse threshold for simple graphs and formulate this threshold as a conjecture.