***Spoiler Free***
This battle of wits is story-based and unmissable, the chance to get this achievement however, is very much so.
This is the last social battle achievement and gained at the very end of the game so don't worry if you haven't gotten it yet. One thing worth bearing in mind is that the solutions to social battles change depending on what you've said to other people throughout the course of the game. I took a very much "Paragon" path in the game, the only exception being that I sided with the "warrior-priest" during the sidequest that awards you the achievement "Talion A.D."
Save regularly after beating the boss that awards you the achievement "The Snake" especially just after meeting a turret mounted on the roof - as far as I remember, it's the only one there.
If you have the Social Interface Augmentation then choose:
Extrapolate > Appeal > (Pheromones) Omega option
If not then choose:
Extrapolate > Appeal > Extrapolate > Extrapolate
If this does not work for you then please let me know, especially if you find a new set of choices and if you know what kind of character you played as so everyone can get this achievement please, cheers guys!
Edit: If you do not have the augmentation and the second option doesn't work then try:
Extrapolate > Extrapolate > Critique > Appeal
Full credit goes to ALMAINYNY for finding this solution