I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality regarding Gromov products and quasi-isometries on a hyperbolic metric space $X$ caught my attention:

For a $C$-quasi-isometric map $f$, $$C^{-1} (x,y)_z-A \leq (f(x), f(y))_{f(z)} \leq C(x,y)_z +A$$ for all $x,y,z \in X$, where $A$ is a constant depending on the quasi-isometry constants of $f$ and the hyperbolicity constant of $X$.

The natural question that came to my mind is whether an analogous inequality might hold if $f$ is just a coarse embedding. It seems that hyperbolicity may not be preserved by coarse embeddings so my guess is that a nice relation like above may not be expected, but can we expect any relation between the Gromov products $(x,y)_z$ and $(f(x), f(y))_{f(z)}$?

I really appreciate any help. Thanks in advance!