Berend–Ernst–Kontorovich–Kumar exceptional-point reflection conjecture

For even $k$, let $X_1,\ldots,X_n$ be $k$-wise independent Bernoulli random variables with common parameter $p\in(0,1)$, and consider the problem of maximizing $\mathbb{P}(X_1=\cdots=X_n=1)$. Writing $S=\sum_{i=1}^n X_i$ and separating the distinguished support point $n$, the conjecture asserts that optimal support configurations at parameter $p$ are carried to optimal support configurations at parameter $1-p$ by the reflection $R(s)=n-1-s$ on the remaining support points. Equivalently, if $A\subseteq\{0,\ldots,n-1\}$ is an optimal support configuration at $p$, then $R(A)=\{n-1-s:s\in A\}$ is an optimal support configuration at $1-p$; exceptional parameter values and the multiplicities of local support changes are reflected correspondingly.