The directed Nine Dragon Tree Conjecture for branchings

Let $D$ be a digraph. Write $\gamma(D)$ for the fractional arboricity of its underlying graph, and let $\Delta^-(D)$ and $\Delta^+(D)$ denote its maximum in-degree and maximum out-degree, respectively. A branching is a spanning subdigraph whose components are arborescences. The directed Nine Dragon Tree Conjecture. For positive integers $k$ and $d$, if

$$\gamma(D) \leq k+\frac{d-k}{d+1},\qquad \Delta^-(D)\leq k+1,$$

then $D$ decomposes into $k+1$ branchings $B_1,\ldots,B_k,B_{k+1}$ such that

$$\Delta^+(B_{k+1})\leq d.$$

This is proposed as the directed analogue of the Nine Dragon Tree Conjecture. The source gives no resolution, so the conjecture remains open.