I am interested in the status of the subgroup membership problem (MP) for finitely presented Noetherian groups. That is, given a finite presentation $\langle X,R\rangle$ for a Noetherian group, \begin{equation*} \begin{array}{ll} \text{Input:} & u,v_1,\ldots,v_n \text{ words in } X^{\pm}\\ \text{Decide:}& u \in \langle v_1,\ldots,v_n \rangle \end{array} \end{equation*}

Of course, relevant to this is the possibility that every finitely presented Noetherian group is virtually polycyclic (and so has solvable MP). Is this still an open ~~conjecture~~ question? Any information on decision problems for Noetherian groups will be also greatly appreciated.