Hopkins's formal low-degree conjecture for symmetric almost-independent distributions

Let $\Omega$ be finite or $\mathbb R$, let $k$ be fixed, and let $N=\binom nk$. Let $\nu$ be a product distribution on $\Omega^N$, let $\mu$ be another distribution on $\Omega^N$, and let $T_\delta\mu$ denote the corresponding noisy or smoothed distribution. Suppose that $\mu$ is $S_n$-invariant and $(\log n)^{1+\Omega(1)}$-wise almost independent with respect to $\nu$. Hopkins's Formal Low-Degree Conjecture. No polynomial-time test distinguishes $T_\delta\mu$ from $\nu$ with probability $1-o(1)$ for any $\delta>0$; formally, for every polynomial-time test there is $\delta'>0$ such that the displayed average error is at most $1-\delta'$. This is a formalization of the low-degree heuristic under symmetry and almost-independence hypotheses.