I've successfully read from & written updated BIOS to this motherboard, both verified. The modelling is a bit confusing, there are a few revisions of the board. I can't say for certain, but it seems G31M3 and G31M3-F are the same board. BIOS images from G31M3-F product page work on my board missing the -F (which I can't find any official mention of).
Attached are the output of flashrom -p internal -V, lspci -nnvvxxx and superiotool -deV in results1.txt, results2.txt and results3.txt respectively. Please tell me if attachments are not acceptable and I will reply with the full text (the outputs are very long!).