Non-isomorphism conjecture for pure infinitely braided Thompson groups

Let $BF_n(\{1\})$ and $BF_n(\mathcal{P}_n)$ be the groups associated respectively with the trivial label group and with the pure braid group $\mathcal{P}_n$ in the construction of pure infinitely braided Thompson groups. Non-isomorphism conjecture.

$$BF_n(\{1\})\not\cong BF_n(\mathcal{P}_n).$$

The preceding discussion gives finite generating sets for both groups, while the paper leaves open whether they are isomorphic and also asks for minimal generating sets and explicit presentations.