Partition of Sparse Graphs into Two Forests with Bounded Degree

Borodin and Kostochka proved that for $d_2\ge 2d_1+2$ and a graph $G$ where every subgraph $H$ satisfies $$e(H)<(2-\frac{d_2+2}{(d_1+2)(d_2+1)})n(H)+\frac{1}{d_2+1}$$ has a vertex partition $V(G)=V_1\cup V_2$ such that $G[V_i]$ has maximum degree at most $d_i$ for each $i$. We show that under the same conditions we can additionally conclude that each $G[V_i]$ is a forest.