Measure-independence conjecture for dimensions of polynomial-growth harmonic functions

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

$$and$$

be courteous measures on $G$ with an exponential tail. Let

$$denote the space of$$

-harmonic functions on $G$ with growth degree at most $k$.

Harmonic-function dimension conjecture. For every $k\geq 0$,

$$\dim \mathsf{HF}_k(G,\mu)=\dim \mathsf{HF}_k(G,\nu).$$

The case $k=0$ concerns bounded harmonic functions and is included explicitly in the source. The conjecture predicts that these dimensions depend on $G$ but not on the chosen courteous measure; the source notes that analogous independence is known for finitely generated groups under the cited results.