Prof. Kapovich: Thank you very much for the overview and the reference to the Fisher-Whyte paper, which I did not know. I have accepted this answer as a clear summary of the current state of knowledge: my question is open. I thank all the experts who took time to comment.

Prof. de Cornulier: Thank you. So it appears that my slightly stronger question may be open as well (or false), but certainly that no proof is known. Do you know of a reference to read more about these specific examples?

Prof. Valette: Thank you for the response. I don't see how Theorem 1.1.2 in Kleiner-Leeb answers my question, as the quasi-isometries considered in that theorem are assumed to be (coarsely) surjective, whereas I am only interested in an embedding result. (Or did I misunderstand? I reiterate my lack of expertise re: buildings.)