Mikhail Gromov states that he "tried for about 10 years to prove that every hyperbolic group is realizable by a space of negative curvature" in his interview with Martin Raussen and Christian Skau (page 6, link below). He says that he "couldn't do it and this is still unknown".

Is there any information available about the current status of this conjecture?

https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/rtx100300391p.pdf