The WMS divergence conjecture for regular LDPC codes

Let $(d_v,d_c)$-regular LDPC codes have girth $\Omega\left(\log n\right)$, and transmit them over a binary symmetric channel with cross-over probability $p$. Let $p^*$ be the bit-error-rate threshold for weighted min-sum decoding with $\beta=\frac{1}{d_v-1}$. WMS divergence conjecture. The WMS decoding diverges to consistent messages with high probability for all $p<p^*$. If true, this would show that the threshold of WMS decoding with $\beta=\frac{1}{d_v-1}$ gives a lower bound for the threshold of LP decoding; the conjecture concerns the relationship between iterative WMS decoding and LP decoding for locally tree-like regular LDPC codes.