We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph H , the H -colouring problem is to decide whether a given graph has a homomorphism to H . By a result of Hell and Nešet?il, this problem is NP-hard for any non-bipartite graph H . In the context of promise constraint satisfaction problems, Brakensiek and Guruswami conjectured that this hardness result extends to promise graph homomorphism as follows: fix any non-bipartite graph H and another graph G with a homomorphism from H to G , it is NP-hard to find a homomorphism to G from a given H -colourable graph. Arguably, the two most important special cases of this conjecture are when H is fixed to be K 3 (and G is any graph with a triangle) and when G = K 3 (and H is any 3-colourable graph). The former case is equivalent to the notoriously difficult approximate graph colouring problem. In this paper, we confirm the Brakensiek-Guruswami conjecture for the latter case. Our proofs rely on a novel combination of the universal-algebraic approach to promise constraint satisfaction, that was recently developed by Bulín and the authors, with some ideas from algebraic topology.