Balanced max 2-sat might not be the hardest

We show that, assuming the Unique Games Conjecture, it is NP-hard to approximate MAX2SAT within αLLZ-+ε, where 0.9401 < αLLZ- < 0.9402 is the believed approximation ratio of the algorithm of Lewin, Livnat and Zwick [28]. This result is surprising considering the fact that balanced instances of MAX2SAT, i.e., instances where each variable occurs positively and negatively equally often, can be approximated within 0.9439. In particular, instances in which roughly 68% of the literals are unnegated variables and 32% are negated appear less amenable to approximation than instances where the ratio is 50% - 50%.