Measure-independence conjecture for finite-dimensional spaces of linear-growth harmonic functions

Let $G$ be a compactly generated locally compact group, and let

be a symmetric, adapted probability measure on $G$ with an exponential tail. Let

be the space of linearly growing

-harmonic functions on $G$. **Finite-dimensionality characterization conjecture.** The group $G$ has polynomial growth if and only if

is finite dimensional.

The conjecture is attributed in the source to the earlier work cited as MY16. The paper's main result establishes this characterization in the connected setting, while the analogous general statement is presented here as a conjecture.