I think the one most likely to react would be AIB. Other banks don't seem willing to offer as low rates as AIB or BOI. Some (Halifix for certain) have increased rates in the face of falling ECB rates.
It's not really a great offer to be honest.
I'd prefer to take my chances on a variable at the same rate 2.65%.
The best way for a bank to get market share for first time buyers will be to relax their lending criteria not through reduced interest rates. It's all well and good offering a low rate but if you are stressing the repayments at 6%, not many first time buyers will qualify for the rate.