Correct-path survival conjecture for split-reduced successive cancellation list decoding

A polar code is decoded using a split-reduced successive cancellation list decoder, with $u_i$ denoting the current unfrozen bit and $P_e(u_i)$ its decoding error probability. Under the Gaussian approximation, consider a correct decoding path that survives through $u_1^{i-1}$.

Correct-path survival conjecture. As $P_e(u_i)$ approaches zero, with high probability the current path survives at $u_i$ without splitting and $u_i$ is correctly decoded. If the subsequent subchannels corresponding to $u_{i+1}^N$ become increasingly reliable, then the correct path survives until termination without splitting with high probability.

This conjecture concerns the path behavior induced by the proposed split-reduced rule; the paper reports empirical evidence for it, while a quantitative result is difficult because list-decoding pruning couples exponentially many error patterns.