The replica-symmetry-breaking transition conjecture for random uniquely extendable CSPs

Let $\pi$ be a distribution over $\Lambda_{k,r}$ and let $H_n(\pi,k,m)$ be the associated random uniquely extendable constraint system. Suppose that either $k\geq 4$ and $\operatorname{supp}(\pi)$ is reducible, or $k=3$ and every function in $F_2(\operatorname{supp}(\pi))$ is symmetric. Let $\varepsilon>0$. Replica-symmetry-breaking conjecture. Below the threshold, if $m<(d_k/k-\varepsilon)n$, and $x$ is a uniform random solution while $u,v$ are independent uniform variables, then $\mathbb P[x_u=\sigma,x_v=\tau]=1/q^2+o(1)$ for all $\sigma,\tau\in\Omega$. Above the threshold, if $m>(d_k/k+\varepsilon)n$ and $Q_\psi\neq\varnothing$, where $Q_\psi$ is the set of constant solutions of $\psi$, then a uniform random solution does not satisfy property $\mathrm{RS}$ a.a.s. This conjectures a replica-symmetry-breaking, or condensation, transition at density $d_k$.