  In this article we study a coarse version of the $K$-theoretic Farrell--Jones
conjecture we call coarse or bounded isomorphism conjecture. Using controlled
category theory we are able to translate this conjecture for asymptotically
faithful covers into a more familiar form. This allows us to prove the
conjecture for box spaces of residually finite groups whose Farrell--Jones
assembly map with coefficients is an isomorphism.
