The rough-isometric embedding obstruction for isotropic trees

Let $R=R((p_n),(q_n))$ and $R'=R((p_n'),(q_n'))$ be isotropic trees. Let ${\mathcal{X}}((p_n),(q_n),(p_n'),(q_n'))$ denote the associated set of obstructions to a rough isometric embedding. Rough-isometric embedding obstruction. If

$${\mathcal{X}}((p_n),(q_n),(p_n'),(q_n'))$$

is unbounded, then there does not exist a rough isometric embedding of $R$ into $R'$. This claim would remove the need for the earlier argument's assumption that $(p_n')$ is constant, and gives a general obstruction to rough-isometric embeddings between isotropic trees.