We consider the complexity of LS$_+$ refutations of unsatisfiable instances of Constraint Satisfaction Problems (CSPs) when the underlying predicate supports a pairwise independent distribution on its satisfying assignments. This is the most general condition on the predicates under which the corresponding MAX-CSP problem is known to be approximation resistant.

We show that for random instances of such CSPs on $n$ variables, even after $\Omega(n)$ rounds of the LS$_+$ hierarchy, the integrality gap remains equal to the approximation ratio achieved by a random assignment. In particular, this also shows that LS$_+$ refutations for such instances require rank $\Omega(n)$. We also show the stronger result that refutations for such instances in the \emph{static} LS$_+$ proof system requires size $\exp(\Omega(n))$.