  We prove the BMM symmetrising trace conjecture for the exceptional complex
reflection groups $G_4,\,G_5,\,G_6,\,G_7,\,G_8$ using a combination of
algorithms programmed in different languages (C++, SAGE, GAP3, Mathematica).
Our proof depends on the choice of a suitable basis for the generic Hecke
algebra associated with each group.
