The Abelian embedding conjecture for approximability of satisfiable $k$-CSPs

Let $mu$ be a distribution on $\Sigma^k$. An Abelian embedding of $mu$ is an Abelian group $G$ and mappings $\sigma_i: \Sigma \rightarrow G$, $1 leqslant i leqslant k$, not all constant, such that

$$\sum_{i=1}^k \sigma_i(a_i)=0_G$$

for every $(a_1,\ldots,a_k)in \mathsf{supp}(\mu)$. Here, Conclusion

is the paper's stated exponential-product high-degree conclusion for the distribution. **Abelian embedding conjecture.** For a distribution $\mu$ on $\Sigma^k$, Conclusion

holds if and only if $\mu$ admits no Abelian embedding.

The claim proposes that the absence of Abelian embeddings is not only necessary, as shown by the character construction preceding the conjecture, but also sufficient for the relevant approximability conclusion. Its resolution is not specified in the supplied text.