Brough's conjectures on poly-context-free groups

A group is poly-context-free if its word problem is a finite intersection of context-free languages. A group is virtually a property if it has a finite-index subgroup with that property, and a virtual direct product of free groups is a group having a finite-index subgroup that is a direct product of finitely generated free groups. Brough's conjectures. The following three assertions are conjectured: (i) a finitely generated group is poly-context-free if and only if it is a finitely generated subgroup of a virtual direct product of free groups; (ii) a finitely generated solvable group is poly-context-free if and only if it is virtually abelian; and (iii) if a group $G$ is poly-context-free, then $G$ does not have arbitrarily large finite subgroups. These conjectures seek structural characterizations and restrictions for poly-context-free groups; the source presents them as open conjectures.