The issue is that the magic values of parameters in DFT-D2 are available in Grimme's paper and implemented in ATK only for the GGA-type functional. How did the author of "Comparison of calculated (DFT-D2 with LDA) and experimentally measured (ARPES)" perform the DF2-D2 calculations with LDA?
Even they are very lucky to get reasonable results in agreement with experiments for their system, but this doesn't make sense.
The LDA usually gives a stronger bonding than the real one. The DFT-D2 with LDA would give much stronger bonding than LDA. So the use of DFT-D2 with LDA may be quite questionable.