Extension conjecture for edge-disjoint branchings in digraphs

Let $D=(V,A)$ be a digraph, let $\kappa$ be an infinite cardinal, and let $\mathcal{B}_i=(V_i,A_i)$ for $i<\kappa$ be edge-disjoint branchings in $D$. Define

$$D\setminus\mathcal{B}=(V,A\setminus\bigcup_{i<\kappa}A_i).$$

Suppose that $D\setminus\mathcal{B}$ contains no forward-infinite paths and that, for every $v\in V$, there is a system of edge-disjoint paths $\{P_i\}_{i<\kappa}$ in $D\setminus\mathcal{B}$ such that $P_i$ goes from $V_i$ to $v$. Branching extension conjecture. The branchings $\mathcal{B}_i$ can be extended, without changing their root sets, to edge-disjoint spanning branchings of $D$. This is an extension problem for infinitely many edge-disjoint branchings under a no-forward-infinite-path condition; the supplied source does not provide evidence resolving the assertion.