Decomposing highly edge-connected graphs into paths of any given length

In 2006, Barát and Thomassen posed the following conjecture: for each tree T, there exists a natural number kT such that, if G is a kT-edge-connected graph and |E(G)| is divisible by |E(T)|, then G admits a decomposition into copies of T. This conjecture was verified for stars, some bistars, paths of length 3, 5, and 2r for every positive integer r. We prove that this conjecture holds for paths of any fixed length.