Decomposing highly connected graphs into paths of length five

Barát and Thomassen (2006) posed the following decomposition 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. In a series of papers, Thomassen verified this conjecture for stars, some bistars, paths of length 3, and paths whose length is a power of 2. We verify this conjecture for paths of length 5.