I recently learned about the following question, asked by I. Kapovich :

Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$, whose Bowditch boundary is homeomorphic to some $n$-sphere ?

One can find the question here, together with a list of questions about boundaries of groups.

However, I was told once that one could realize any finitely generated nilpotent group as the stabilizer of a cusp in a finite volume manifold of pinched negative curvature. This result would follow from

Gromov-Ruh theorem that says that any infranilmanifold is almost flat (the theorem actually states that this is an if and only if condition).

Corollary 6 in P. Ontaneda's paper Pinched smooth hyperbolization (arXiv link here). It is proved there that for any almost flat manifold $Q$, $Q\bigsqcup Q$ bounds geometrically a pinched negatively curved manifold

See also Theorem 1.1 in I. Belegradek and V. Kapovich's paper Classification of negatively pinched manifolds with amenable fundamental groups (Acta Math. Volume 196 (2006), 229-260, arXiv link here). It is proved there that any infranilmanifold can be realized as a horosphere quotient.

Now, if $M$ is a manifold with pinched negative curvature, then its fundamental group $\pi_1(M)$ is hyperbolic relative to the stabilizers of the cusps and the Bowditch boundary is the limit set of $\pi_1(M)$ in $\tilde{M}$, so if $M$ has finite volume, the Bowdtich boundary of $\pi_1(M)$ is a $n$-sphere.

Of course, it could be the case that the question was asked before those papers were written, but

Could someone clarify this ?

What is the current status of this conjecture ?