Soft-information bounds for monotone-function code ensembles

Let $\alpha=C/R$, and let $q^{\mathrm{BEC}}_t$ and $q^{\mathrm{BSC}}_t$ be the dynamical systems initialized at $x_0$ by

$$q^{\mathrm{BEC}}_{t+1}(x_0)=1-\mathcal{H}_{\le d}^{\mathrm{BEC}}(\alpha,q^{\mathrm{BEC}}_t),$$

and

$$q^{\mathrm{BSC}}_{t+1}(x_0)=h^{-1}(\mathcal{H}_{\le d}^{\mathrm{BSC}}(\alpha,q^{\mathrm{BEC}}_t)).$$

Here $\iota^{\mathrm{BP}}_\ell$ is the soft information output after $\ell$ iterations of belief propagation, $h^{-1}$ is the inverse of the binary entropy function on the relevant domain, and $o(1)\to0$ as $k\to\infty$. Soft-information bound conjecture. For a code ensemble generated by a monotone function,

$$q^{\mathrm{BEC}}_\ell(0)+o(1)\ge \iota^{\mathrm{BP}}_\ell\ge 1-h_b\bigl(q^{\mathrm{BSC}}_t(1/2)\bigr)+o(1).$$

The source presents this as a conjectured bound for majority codes, or perhaps more generally for binary monotone functions. Since the supplied parser status is unknown, its resolution is not established here.