Convergence conjecture for adaptive LP decoding

Let a random parity-check code have length $n$, $m=n(1-R)$ parity checks, and arbitrary degree distributions. The adaptive LP decoding algorithm is the iterative algorithm that adds violated parity-check constraints during LP decoding. As $n$ and $m$ increase, Convergence conjecture. the algorithm converges with probability arbitrarily close to $1$ in at most $\alpha$ iterations and with at most an average of $\beta$ final parity-check constraints per check node, where $\alpha$ and $\beta$ are constants independent of the length, rate, and degree distribution of the code. The conjecture predicts size-independent practical complexity for adaptive LP decoding, whereas the preceding discussion establishes only empirical evidence from simulations; its resolution is not given here.