Gao–Yang's directed Nine Dragon Tree conjecture

Let $D$ be a digraph. A branching is a digraph whose components are arborescences, and write $\gamma(D)$ for the directed fractional packing parameter and $\Delta^-$ and $\Delta^+$ for minimum and maximum indegree and outdegree, respectively. Let $k,d$ be positive integers. Gao–Yang's directed Nine Dragon Tree conjecture. If

$$\gamma(D)\leq k+\frac{d-k}{d+1}$$

and

$$\Delta^-(D)\leq k+1,$$

then $D$ decomposes into $k+1$ branchings $F_1,F_2,\ldots,F_{k+1}$ with

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

This is the digraphic version of the Nine Dragon Tree conjecture introduced by Gao and Yang; the supplied text says that only part of it had been proved, so its full status remains open.