Rich $2$-to-$1$ Games conjecture

Let $\Psi=(L\cup R,E,\Sigma_L,\Sigma_R,\Phi)$ be a Rich $2$-to-$1$ Games instance with $|\Sigma_L|=n$ and $|\Sigma_R|=n/2$. For each $u\in L$, the distribution of partitions of $\Sigma_L$ induced by uniformly sampled incident constraints is uniform over all partitions of $\Sigma_L$ into pairs. Let ${\sf Gap\text{-}Rich\text{-}2\text{-to-}1}_n[1,\delta]$ denote the promise problem distinguishing instances with an assignment satisfying all constraints from instances in which every assignment satisfies at most a $\delta$ fraction of the constraints. Rich $2$-to-$1$ Games conjecture. For every $\delta>0$, there is a sufficiently large even $n\in\mathbb{N}$ such that ${\sf Gap\text{-}Rich\text{-}2\text{-to-}1}_n[1,\delta]$ is NP-hard. This stronger variant was proposed to support hardness results for satisfiable instances and to provide additional structure beyond ordinary $2$-to-$1$ Games; the supplied text gives no resolution status.