The threshold conjecture for distributions of reducible uniquely extendable constraints

Let $k\geq 3$, and let $\mu$ be a distribution over the set $\Lambda_{k,r}$ of $k$-ary uniquely extendable constraint functions. Suppose that, if $k\geq 4$, every function in $\operatorname{supp}(\mu)$ is commutative and reducible, or, if $k=3$, every function $\psi\in\operatorname{supp}(\mu)$ is commutative with symmetric $f_{\psi}^{(2)}$. Let $d_k$ be the random $k$-XORSAT threshold constant. Threshold conjecture. If $\mu$ admits no constant solutions, then $d_k/k$ is the satisfiability threshold, in terms of the density $d=m/n$, of $H_n(\mu,k,m)$. This conjecture generalizes the fixed-constraint threshold result to random choices from a distribution of constraint functions.