Theorem. Let $\phi:X\rightarrow Y$ be a quasi-isometry between two (Gromov) hyperbolic spaces $X$ and $Y$. If $X$ and $Y$ are proper, then $\phi$ induces a homeomorphism between their boundaries.
The proof of the above statement is well-written in Bridson and Haefliger's book.
My question is that `can we drop the condition that $X$ and $Y$ are proper?'. In some papers about boundaries of hyperbolic spaces, the authors usually say that the above theorem is true without mentioning that $X$ and $Y$ are proper. If you know the answer or any references, then let me know.