Forest-and-bounded-diameter-forest decomposition conjecture

Let $d\geq1$ be a natural number, and let $G$ be a graph that is the union of a forest and a second forest whose components have diameter at most $d$.

Forest-and-bounded-diameter-forest decomposition conjecture. There exists a natural number $f(d)$ such that $G$ can be partitioned into two forests, each of whose components has diameter at most $f(d)$.

This conjecture generalizes the paper's forest-and-star-forest result. The paper confirms it when $d\leq2$, with $f(2)\leq18$, but the assertion for general $d$ remains open.