Induced homeomorphism from a quasi-isometry between hyperbolic spaces

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.