UGC hardness of the predicate $NAE(x_1, ..., x_\ell)$ for $x_i \in GF(k)$? Background:

In Subhash Khot's original UGC paper (PDF), he proves the UG-hardness of deciding whether a given CSP instance with constraints all of the form Not-all-equal(a, b, c) over a ternary alphabet admits an assignment satisfying 1-$\epsilon$ of the constraints or whether there exist no assignments satisying $\frac{8}{9}+\epsilon$ of the constraints, for arbitrarily small $\epsilon > 0$.

I'm curious whether this result has been generalized for any combination of $\ell$-ary constraints for $\ell \ge 3$ and variable domains of size $k \ge 3$ where $\ell \ne k \ne 3$. That is,

Question:


Are there any known hardness of approximation results for the predicate $NAE(x_1, \dots, x_\ell)$ for $x_i \in GF(k)$ for $\ell, k \ge 3$ and $\ell \ne k \ne 3$?


I'm especially interested in the combination of values $\ell = k$; e.g., the predicate Not-all-equal($x_1, \dots, x_k$) for $x_1 \dots, x_k \in GF(k)$.
I landed on this page from a search about NAE-3SAT.

I am pretty sure that for the problem you are asking, it should be NP-hard to tell if the instance is satisfiable, or if at most $1-1/k^{\ell-1}+\epsilon$ fraction of constraints can be satisfied. That is, a tight hardness result (matching what simply picking a random assignment would achieve), for satisfiable instances, and no need for the UGC.

For $k=2$ and $\ell \ge 4$, this follows from Hastad's factor 7/8+epsilon inapproximability result for 4-set-splitting (which can then be reduced to k-set splitting for $k > 4$). If negations are okay, one can also use his tight hardness result for Max ($\ell-1$)-SAT.

For $k=\ell=3$, Khot proved this in a FOCS 2002 paper "Hardness of coloring 3-colorable 3-uniform hypergraphs." (That is, he removed the original UGC assumption.)

For $\ell=3$ and arbitrary $k\ge 3$, Engebretsen and I proved such a result in "Is constraint satisfaction over two variables always easy? Random Struct. Algorithms 25(2): 150-178 (2004)". However, I think our result required "folding" i.e., the constraints will actually be of the form NAE($x_i+a,x_j+b,x_k$) for some constants $a,b$. (This is the analog of allowing negations of Boolean variables.) 

For the general case, I don't know if this has been written down anywhere. But if you really need it, I can probably find something or check the claim.