Difference between high-energy photon index of the GBM emission (β_BEST) and the LLE photon index (α_LLE) as a function of the spectral lag between Band 1 (GBM-NaI) and Band 4 (LAT-LLE) for 35 out of 70 GRBs in our sample. Short GRBs are indicated by stars. Green points represent GRBs that deviate from the expected trend, corresponding to GRBs 100724B, 100826A and 131108A. The dashed-blue line represent the arbitrary threshold between hard and soft spectra in the framework of the phenomenological scenario described in Section 7. Spectral lags τ are reported in symmetric-logarithmic scale for visualization purpose.