Finitely presented BVC groups are virtually cyclic

Let $G$ be a group. It has BVC if there are virtually cyclic subgroups $V_1,\ldots,V_n$ such that every virtually cyclic subgroup of $G$ is conjugate to a subgroup of some $V_i$. BVC conjecture. If $G$ is finitely presented and has BVC, then $G$ is virtually cyclic. The paper proves the corresponding assertion for several classes of finitely generated groups and presents this finitely presented version as a conjecture that may hold in general.