Constant-iteration and constraint conjecture for adaptive LP decoding

Let a random parity-check code have length $n$, rate $R$, and $m=n(1-R)$ parity checks, with an arbitrary degree distribution. Consider the adaptive LP decoding algorithm, and let $b1$ and $b2$ denote constants.

Adaptive LP decoding conjecture. As $n$ increases, the algorithm converges with probability arbitrarily close to $1$ in at most $b1$ iterations and with at most $b2$ final parity-check constraints per check node, where $b1$ and $b2$ are independent of the code's length, rate, and degree distribution.

The conjecture formalizes the observed practical speed of adaptive LP decoding: simulations suggest substantially fewer iterations and final parity-check constraints than the general worst-case guarantee. Its validity for random parity-check codes with arbitrary degree distributions remains open.