Connamacher and Molloy's satisfiability-threshold conjecture for random uniquely extendable CSPs

Let $k\frac{}{}$ and $r$ be fixed integers with $k\geq 3$ and $r\geq 2$. Let $d_k$ be the constant defined in the random $k$-XORSAT satisfiability-threshold theorem. Let $\Phi$ be a random $(k,r)$-UE-SAT instance with $n$ variables and $m$ $k$-ary uniquely extendable constraints. Connamacher and Molloy's conjecture. For every $\varepsilon>0$, a.a.s. $\Phi$ is satisfiable if $m<(d_k/k-\varepsilon)n$, and a.a.s. $\Phi$ is unsatisfiable if $m>(d_k/k+\varepsilon)n$. This conjecture proposes that the random $(k,r)$-UE-SAT threshold agrees with the random $k$-XORSAT threshold.