Biharmonic maps are the solutions of a variational problem, but they are difficult to study with variational methods, in part due to the lack of coercivity of the underlying functional. Recently Hornung was able to apply the direct method to a modified functional under the assumption that the dimension of the domain is 3 or 4. In this paper, the corresponding minimisers are studied in the case of a homogeneous target space. It is shown that they also represent minimisers of the original functional among a suitable class of comparison maps. Moreover, they solve the corresponding Euler-Lagrange equation if it is interpreted in a sufficiently weak sense.