Montassier et al.'s bounded-degree forest decomposition conjecture

Let $k$ and $d$ be positive integers, and let $G$ be a graph. Write $\Upsilon_f(G)$ for the fractional arboricity of $G$, defined by

$$\Upsilon_f(G)=\max_{H\subseteq G}\frac{|E(H)|}{|V(H)|-1}.$$

Montassier et al.'s conjecture. If

$$\Upsilon_f(G)\leq k+\frac{d}{k+d+1},$$

then $E(G)$ can be decomposed into $k+1$ forests, one of which has maximum degree at most $d$.

This conjecture generalizes the known decompositions of graphs with fractional arboricity at most $\tfrac{4}{3}$ or $\tfrac{3}{2}$. It is open for $k\geq 2$ or $d\geq 7$.