The weakened-support conjecture for random uniquely extendable constraints

Let $\pi$ be a distribution over $\Lambda_{k,r}$, and let $H_n(\pi,k,m)$ be the associated random uniquely extendable constraint system. Weakened-support conjecture. Theorem~ should hold if either $k\geq 4$ and every $\psi\in\operatorname{supp}(\pi)$ is reducible, or $k=3$ and $f_{\psi}^{(2)}$ is symmetric for every $\psi\in\operatorname{supp}(\pi)$. This weakens the condition that the entire support is reducible and is motivated by the fact that symmetry of each individual $f_{\psi}^{(2)}$ need not imply symmetry of $F_2(\Psi)$.