The domino problem for hyperbolic groups

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.