The combinatorial low-degree correlation conjecture for pairwise-connected distributions

Let $\Sigma$ be a finite alphabet, let $k\in\mathbb{N}$, and let $\mu$ be a pairwise-connected distribution over $\Sigma^k$ in which every atom has mass at least $\alpha$. A function $P\colon\Sigma^n\to\mathbb{C}$ is a combinatorial degree-$k'$ function if it factors as

$$P(x)=\prod_{T\in\binom{[n]}{k'}}P_T(x_T)$$

for suitable functions $P_T\colon\Sigma^T\to\mathbb{C}$. The combinatorial low-degree correlation conjecture. For every $k\in\mathbb{N}$ there exists $k'\in\mathbb{N}$ such that, for every $\varepsilon,\alpha>0$, there are $d\in\mathbb{N}$ and $\delta>0$ with the following property: if $f_1,\ldots,f_k\colon\Sigma^n\to\mathbb{C}$ are $1$-bounded and

$$\left|\mathbb{E}_{(x_1,\ldots,x_k)\sim\mu^{\otimes n}}[f_1(x_1)\cdots f_k(x_k)]\right|\geq\varepsilon,$$

then there exist a combinatorial degree-$k'$ function $P\colon\Sigma^n\to\mathbb{C}$ with $2$-norm equal to $1$, and a function $L\colon\Sigma^n\to\mathbb{C}$ with $2$-norm equal to $1$ and ${\sf \deg}(L)\leq d$, such that

$$\left|\langle f_1,L\cdot P\rangle\right|\geq\delta.$$

The preceding results settle the corresponding question for arity three and for distributions with no Abelian embeddings, but the general case for $k\geq4$ is open; this conjecture is presented as a natural and plausible answer in the Abelian-embedding case.