Existence of a primitive ergodic element in a treeable equivalence relation

Let ${\mathcal{R}}$ be an ergodic treeable equivalence relation of cost $>1$, and let $[\mathcal{R}]$ denote its full group. A subequivalence relation of ${\mathcal{R}}$ is primitive if it admits a treeing that extends to a treeing of ${\mathcal{R}}$.

Primitive-element conjecture. There exists an ergodic element $f \in [\mathcal{R}]$ such that the subequivalence relation generated by $f$ is primitive in ${\mathcal{R}}$.

Such an element would provide a primitive ergodic subequivalence relation generated by a single transformation, complementing the paper's results on surjections from treeable equivalence relations. The supplied context does not state whether this conjecture has been resolved.