The dichotomy conjecture for conjugacy of discrete subgroups of connected Lie groups

Let $G$ be a connected Lie group, let $ath.E(G,\mathcal{D}(G))$ denote the conjugacy relation on the space $\mathcal{D}(G)$ of discrete subgroups of $G$, and call an equivalence relation essentially hyperfinite or essentially countable universal in the usual Borel-reducibility sense. Connected-Lie-group conjugacy dichotomy. Exactly one of the following holds:

1. $E(G,\mathcal{D}(G))$ is essentially hyperfinite; or
2. $E(G,\mathcal{D}(G))$ is essentially countable universal.

The proposed dichotomy is motivated by the result that a connected Lie group fails to contain a discrete nonabelian free subgroup exactly when it is a compact extension of a solvable subgroup. In particular, the conjecture predicts essential hyperfiniteness for conjugacy on discrete subgroups of compact-by-solvable Lie groups; the general dichotomy remains open.