thank you for your interesting question. The question of whether a particular ligand occurs as a conducting ligand in a particular MOF cannot be answered in a general way. It depends on the size of the band gap. For example, in MOF-5 [Zn4O(BDC)3], the band gap is 4.6 eV. BDC acts as an insulator in this case. The following review article may be helpful to you:
Dear Florian Morsbach, thank you very much for your prompt response. I have gone through the above review. But, no information regarding the conducting nature of BDC. Moreover, all MOFs containing BDC are least conducting ranging the conductivity in the order 10^-8 and less. That kept in ambiguity. Even though having a continuous pi-cloud, why BDC is a poorly conducting. Of course, bandgap certainly plays a major role, is there any way to tune it for better conduction especially with BDC.