Olshanskii–Osin–Sapir conjecture on Floyd boundaries and relative hyperbolicity

Let $G$ be a finitely generated group. Its Floyd boundary is the boundary obtained from a Floyd metric associated to a Floyd function. A group is hyperbolic relative to a collection of subgroups if it admits the corresponding relatively hyperbolic structure.

Olshanskii–Osin–Sapir conjecture. If $G$ has non-trivial Floyd boundary, then $G$ is hyperbolic relative to a collection of proper subgroups.

The converse is stated in the source to follow from Gerasimov's theorem on Floyd maps for relatively hyperbolic groups. The conjecture concerns whether every finitely generated group with non-trivial Floyd boundary must arise from a relative hyperbolic structure.