Confusion in 2012 paper by Austrin and Håstad regarding hardness of approximating GLST The paper in question is "On the Usefulness of Predicates", Per Austrin, Johan Håstad (arXiv:1204.5662 [cs.CC]).  

On page 13, Example 8.2 they define a predicate $P$ which is $GLST$ with an additional accepting predicate of all $1$'s. The claim is that this predicate can be shown approximation resistant with Theorem 8.3, which requires that $P$ accept all strings $x_1 x_2 x_3 x_4$ such that $\prod_1 ^3 x_i = -1$ and $x_3 = -x_4$.  

In particular, $P$ should accept $(1,1,-1,1)$ but the definition of $GLST$ provided requires that $x_2 \ne x_4$.
Following Emil's suggestion and using trial and error, define $P'(x_1, x_2, x_3, x_4) \equiv P(-x_1, x_2, x_3, x_4)$

Instances of $\text{CSP}(P')$ are isomorphic to instances of $\text{CSP}(P)$, via adding a negation in every constraint where $x_1$ appears. Then we apply the theorem on $P'$ after checking that the $\{3,4\}$ fourier coefficient is still positive. We can conclude $P'$ is resistant and therefore $P$ is as well.