  Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to
\{0,1\}^k$ is said to be a direct product function if there exists a function
$f\colon X(1)\to \{0,1\}$ such that $F(\sigma) = (f(\sigma_1), \ldots,
f(\sigma_k))$ for each $k$-face $\sigma$. In an effort to simplify components
of the PCP theorem, Goldreich and Safra introduced the problem of direct
product testing, which asks whether one can test if $F\colon X(k)\to \{0,1\}^k$
is correlated with a direct product function by querying $F$ on only $2$
inputs. Dinur and Kaufman conjectured that there exist bounded degree complexes
with a direct product test in the small soundness regime. We resolve their
conjecture by showing that for all $\delta>0$, there exists a family of
high-dimensional expanders with degree $O_{\delta}(1)$ and a $2$-query direct
product tester with soundness $\delta$.
  We use the characterization given by a subset of the authors and
independently by Dikstein and Dinur, who showed that some form of non-Abelian
coboundary expansion (which they called "Unique-Games coboundary expansion") is
a necessary and sufficient condition for a complex to admit such direct product
testers. Our main technical contribution is a general technique for showing
coboundary expansion of complexes with coefficients in a non-Abelian group.
This allows us to prove that the high dimensional expanders constructed by
Chapman and Lubotzky satisfies the necessary conditions, thus admitting a
2-query direct product tester with small soundness.
