Is there any known example of a one-ended finitely presented group with exponential growth that does not contain a quasi-isometric copy of the hyperbolic plane?

This question is motivated by the following question of Papasoglu mentioned in the paper *'Quasi-hyperbolic planes in hyperbolic group'* by Bonk–Kleiner which asks whether every one-ended finitely presented group $G$ contains a quasi plane- the image of a uniform embedding $P\rightarrow G$ where $P$ is a complete Riemannian plane with bounded geometry.

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