While discussing with a colleague of a possible use of lacunary hyperbolic elementarily amenable groups introduced by Olshanskii, Osin & Sapir, it occurred to me that I was not aware whether these groups (or any amenable lacunary hyperbolic group) could be Liouville (in the sense that there are no non-constant harmonic functions on some Cayley graph). Hence the question in the title:

**Question:** Are there lacunary hyperbolic groups which admit a Cayley graph so that there is no non-constant harmonic function on that graph?

Or more restrictively, can this be said of the groups introduced by Olshanskii, Osin & Sapir (Lacunary hyperbolic groups, section 3.5)?