Sparse graph decomposition conjecture with no overfull set

Let $G$ be a graph and let $k,d\in\mathbb{N}$. A vertex subset $A\subseteq V(G)$ is overfull if

$$|E(G[A])|>(k+1)(|A|-1).$$

A graph is $(k,d)$-sparse when it satisfies the corresponding $(k,d)$-sparseness inequality, and it is $(k,d)$-decomposable if its edges can be decomposed into $k$ forests and one $d$-bounded forest. Sparse graph decomposition conjecture. If $G$ is $(k,d)$-sparse and has no overfull set, then $G$ is $(k,d)$-decomposable. This conjecture strengthens the Nine Dragon Tree Conjecture because the fractional-arboricity hypothesis implies both $(k,d)$-sparseness and the absence of overfull sets. The supplied text does not state that this stronger conjecture has been resolved.