Finite procyclic covering criterion for commutators

Let $G$ be a profinite group whose commutator subgroup is finite-by-procyclic. For a subgroup $K$, write $\pi(K)$ for the set of primes dividing the orders of elements of $K$. Finite procyclic covering conjecture. The commutators in $G$ are covered by finitely many procyclic subgroups if and only if $G'$ is a product of a normal finite subgroup $M$ and a procyclic subgroup $H$ such that

$$\pi(M)\cap\pi(H)=\emptyset.$$

This refines the known characterization of countable procyclic coverings and the finite-covering results for profinite and pro-$p$ groups; the stated criterion is presented as a conjecture for the profinite case.