The entropy noise sensitivity characterization conjecture

Let $(G,\mu)$ be a random walk, where $G$ is a group and $\mu$ is its probability measure. The walk is entropy noise sensitive when its entropy noise sensitivity is defined in the sense of the paper, and it is Liouville when its Poisson boundary is trivial. Entropy noise sensitivity characterization conjecture. A random walk $(G,\mu)$ is entropy noise sensitive if and only if it is Liouville. The paper establishes that entropy noise sensitivity implies the Liouville property, while the converse is presented as a conjecture and is not resolved here.