The Balanced Max-2-SAT is a special case of Max-2-SAT (each clause is a disjunction of exactly 2 literals) in which for every variable $x$, there is a $k$ such that $x$ appears positive exactly $k$ times in the clauses, and exactly $k$ times negative in the clauses.
I'm looking for a published proof of NP-Hardness for this problem, for references.
It is implied NP-Hard by [this paper], but I can't find the original NP-Hardness proof.
Is it a consequence of the hardness of Max-2-SAT ?
Also as a bonus, Balanced Max-2-SAT($k$) is the subcase in which every variable appears exactly $k$ times positive, exactly $k$ times negative. Is there some known $k$ for which NP-Hardness is published? I know that Balanced Max-3-SAT(2) is hard, but found nothing for the $2$ case.