The Baby Remeslennikov Conjecture

Let $G$ be the group and $a$, $d$, and $P$ the element, rank, and set of primes occurring in conditions (a), (b), and (c) of the surrounding discussion. Baby Remeslennikov Conjecture. The implication (c) $\Rightarrow$ (a) holds. This conjecture concerns the proposed equivalence of the stated algebraic and profinite conditions; the supplied text does not indicate whether it has been resolved.