The large flame extension conjecture for rooted digraphs

Let $D$ be an $r$-rooted digraph, and let a flame be a subdigraph satisfying the flame condition defined in the paper; a flame is large when it satisfies the corresponding largeness condition. An $r$-rooted digraph is a digraph rooted at $r$ in the sense of the paper.

Large flame extension conjecture. In every $r$-rooted digraph $D$, every flame extends to a large flame. In particular, every $r$-rooted digraph $D$ admits a large flame.

The preceding theorem proves this when the digraph is acyclic and has no backward-infinite paths. The conjecture asks whether the restriction on the digraph can be removed.