The nilpotent-by-finite characterization via polynomial harmonic functions

Let $G$ be a connected locally compact and compactly generated group, and let $\mu$ be a courteous measure. Let $HF_k(G,\mu)$ denote the space of $\mu$-harmonic functions with polynomial growth of degree at most $k$. A function is a polynomial of degree at most $k$ with respect to a subgroup $H$ when it has that degree with respect to the polynomial structure induced by $H$.

Nilpotent-by-finite characterization. The group $G$ has a finite-index nilpotent subgroup if and only if $\dim HF_k(G,\mu)<\infty$ and there exists a finite-index subgroup $H$ of $G$ such that every $f\in HF_k(G,\mu)$ is a polynomial of degree at most $k$ with respect to $H$.

The conjecture aims to extend the finitely generated structure theorem to connected locally compact groups. The supplied context proves the inclusion $HF_k(G,\mu)\subset P^k(G)$ when $G$ is connected and nilpotent, but does not establish the converse characterization.