Bounded diameter arboricity plus-one conjecture

Let $G$ be a finite simple graph, let $\Upsilon(G)$ denote its arboricity, and let $\Upsilon_{bd}(\mathcal{G})$ denote the bounded diameter arboricity of a graph class $\mathcal{G}$. For $k\geq 1$, write $\mathcal{A}_k$ for the class of graphs with arboricity at most $k$.

Bounded diameter arboricity plus-one conjecture. The class $\mathcal{A}_k$ has bounded diameter arboricity $k+1$, that is,

$$\Upsilon_{bd}(\mathcal{A}_k)=k+1.$$

The conjecture asks whether allowing one more forest always suffices to partition the edges of every graph of arboricity at most $k$ into forests with uniformly bounded component diameters. The paper proves it for $k\in\{2,3\}$, while the general case remains open.