Hi all! Same story with my late 2011 imac and the 2Gb 6970M. Took it to an Apple service and told me it had to be replaced, about 800euros. Talked to Apple who refused to replace it through the program according to, first, it was bought in 2011 (more than 4 years ago) and later a superior told me that my imac was from an specific production line that had not the problem and thus not fitting the replacement prog. Though (sigh) my card is dead and I find no reasonable replacement in ebay or similar. Took it to a non official service to do the reballing but it didn't work, it was deformed and not fitting anymore.
Shall you get together to do something with apple or a change.org or something, please let me know! From Spain