The converse to Kleiner's theorem for harmonic functions on groups

Let $\mathbb{G}$ be a finitely generated group, and let $\mu$ be a symmetric probability measure on $\mathbb{G}$ with finite support that generates $\mathbb{G}$. Let $\mathsf{HF}_k(\mathbb{G},\mu)$ denote the space of $\mu$-harmonic functions on $\mathbb{G}$ whose growth is bounded by a degree-$k$ polynomial. The converse to Kleiner's theorem. The following are equivalent:

$$\mathbb{G}\text{ is virtually nilpotent};$$ $$\mathbb{G}\text{ has polynomial growth};$$ $$\dim \mathsf{HF}_k(\mathbb{G},\mu)<\infty\text{ for all }k;$$ $$\text{there exists }k\geq 1\text{ such that }\dim \mathsf{HF}_k(\mathbb{G},\mu)<\infty.$$

Kleiner proved the implication from polynomial growth to finite dimensionality of spaces of polynomial-growth harmonic functions; the converse would characterize virtually nilpotent groups through these spaces.