BP-collapse conjecture for good codes

Let ${\mathcal{C}}$ be a code drawn from a sequence of “good” codes, and let ${\bf Y}_2$ be the channel output at the relay. Let ${\bf Y}^{\textrm{BP}}_2$ be the output of belief propagation at the relay. Let $\hat{\bf Y}_2$ and ${\bf \hat Y}^{\textrm{BP}}_2$ be the corresponding quantized outputs, with their components defined by the source's equations for $\hat{\bf Y}_2$ and ${\bf \hat Y}^{\textrm{BP}}_2$. BP-collapse conjecture. For the asymptotic performance of soft-DF-BP, assuming that ${\mathcal{C}}$ is taken from a sequence of “good” codes, an analysis may assume

$${\bf Y}^{\textrm{BP}}_2={\bf Y}_2.$$

Similarly, it may assume

$${\bf \hat Y}^{\textrm{BP}}_2=\hat{\bf Y}_2.$$

This conjecture asserts that BP estimation yields no asymptotic loss relative to using the channel output directly in this setting; the source gives no resolution.