The soluble poly-context-free group conjecture

A group is poly-context-free if its word problem is an intersection of finitely many context-free languages. A group is virtually abelian if it has an abelian subgroup of finite index.

Soluble poly-context-free group conjecture. Every finitely generated soluble group is poly-context-free if and only if it is virtually abelian.

This is the soluble specialization of the general classification conjecture. The source proves that a finitely generated poly-context-free soluble group is virtually abelian except possibly for a case involving an infinite normal torsion subgroup, so the conjecture remains open there.