Countability-free vertex-flame conjecture

The paper considers a rooted digraph $D$ with root $r$, and for each vertex $v\in V(D)-r$ the cardinality $\kappa_D(r,v)$ measures the maximum size of a family of pairwise vertex-disjoint directed paths from $r$ to $v$. Theorem asserts the relevant vertex-flame conclusion under the assumption that $D$ is countable.

Countability-free vertex-flame conjecture. We may omit the countability of $D$ in Theorem.

The preceding discussion notes that the countability assumption can already be weakened to the condition $\kappa_D(r,v)\leq\aleph_0$ for every $v\in V-r$, using Davies-trees. The conjecture asks whether the theorem remains valid without any countability assumption.