It is conjectured that for a discrete, finitely presented group $G$ such that $BG$ satisfies Poincaré duality, there actually exists a closed manifold $M$ which is homotopy equivalent to $BG$. 


This is somehow pointing in the opposite direction as Borel's conjecture, which implies that the homeomorphism type of such a manifold $M$ is uniquely determined. 


Who conjectured this first? Is it also due to Borel, or was it Wall, or somebody else?



Every PD group is $\pi_1$ of an aspherical manifold
The reference for the first appearance of the conjecture (still without the condition that the PD group has to be a priori finitely presented) seems to be <http://www.worldcat.org/title/homological-group-theory-proceedings-of-a-symposium-held-at-durham-in-september-1977-on-homological-and-combinatorial-techniques-in-group-theory/oclc/6022486> from 1977.


You find a survey on the state of the subject around 2000 in [Davis: "Poincaré duality groups"](https://people.math.osu.edu/davis.12/pdgroup.pdf).


