Asymptotic strength of random regular Tanner graphs

Let $d_v$ and $d_c$ be integers with $d_c>d_v\geq 3$. A Tanner graph is asymptotically strong if, for every constant $\beta>0$, there is a constant $\alpha>0$ such that its LP decoder succeeds on the asymmetric LLR vector $\gamma(y,\beta)$ for every $y\in\{0,1\}^n$ of weight at most $\alpha n$. Asymptotic-strength conjecture. For all $d_c>d_v\geq 3$, a random $(d_v,d_c)$-regular Tanner graph is asymptotically strong with high probability. This would extend the known implication from expansion to asymptotic strength to all random regular ensembles, without the integrality restrictions on the expansion parameters used in the preceding theorem.