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 allopwing 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.

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.