Unique fixed-point convergence for monotone-function code ensembles

Let $q^{\mathrm{BEC}}_\ell(x)$ denote the state after $\ell$ iterations of the BEC dynamical system for a code ensemble generated by a monotone function, initialized at $x$. Unique fixed-point convergence conjecture. For any ensemble generated by a monotone function, $q^{\mathrm{BEC}}_\ell(x)$ converges to a unique fixed point independent of $x$. The conjecture is motivated by numerical observations for LDMC(5) and by the corresponding large-degree error dynamics; its general status is not resolved in the supplied source.