DESFIRE does not reply to REQA from MFRC522
It would seem that the antenna matching filter is improperly designed, resulting in the transfer of insufficient power to boot a DESFIRE card from one of these cheap eBay modules.

According to the links I posted above, it seems that a few people have successfully modified the cheap MFRC522 boards to successfully communicate with DESFIRE cards, by replacing capacitors and/or inductors in the filter.  I haven't verified that for myself.