  We prove, for every non-virtually free hyperbolic group $G$, that there is no
algorithm that, given a finite collection of dominoes, determines whether the
Cayley graph of $G$ may be edge-covered by these dominoes so that colours match
at vertices. This answers a conjecture by Aubrun, Barbieri and Moutot and goes
towards settling a long-standing conjecture of Ballier and Stein.
