Thom's conjecture on discrete free subgroups in the hyperfinite full group

Let $\mathcal R$ be the hyperfinite ergodic probability-measure-preserving equivalence relation, and let $[\mathcal R]$ denote its full group. A discrete free subgroup of rank $2$ does not exist in $[\mathcal R]$.

Thom's conjecture. There are no discrete free subgroups of rank $2$ in the full group of the hyperfinite ergodic equivalence relation.

This concerns the distinction between dense free subgroups and discrete free subgroups in full groups of probability-measure-preserving equivalence relations. The source presents the assertion as a conjecture of Thom; no resolution is given here.