The converse Liouville property conjecture for amenable measured groupoids

A measured groupoid is a groupoid equipped with a quasi-invariant measure on its unit space and a compatible measure system on its fibers. It is amenable when it admits an invariant mean, and Liouville when its associated measured random walks have only constant bounded harmonic functions.

Converse Liouville property conjecture. Any amenable measured groupoid is Liouville.

The paper proves the converse implication, that every Liouville measured groupoid is amenable, and observes that the reverse implication is plausible. The converse is therefore presented as an open conjecture in this source.