The harmonic-function characterization of polynomial growth

Let $\mathcal{G}$ be a locally compact compactly generated group, and let $\mu$ be a courteous measure. Let $HF_k(\mathcal{G},\mu)$ denote the space of $\mu$-harmonic functions with polynomial growth of degree at most $k$. In the finitely generated case, a finite-index subgroup means a subgroup of finite index in $\mathcal{G}$.

Harmonic-function characterization. The following conditions are equivalent:

1. $\mathcal{G}$ has polynomial growth.
2. $\dim HF_k(\mathcal{G},\mu)<\infty$ for all $k\geq 1$.
3. $\dim HF_k(\mathcal{G},\mu)<\infty$ for some $k\geq 1$.

In the finitely generated case, these are also equivalent to:

4. $\mathcal{G}$ has a finite-index nilpotent subgroup.

This conjecture seeks a converse to Kleiner's finite-dimensionality theorem and extends the corresponding result known for finitely generated solvable groups to locally compact compactly generated groups.