Lipschitz-to-linear harmonic-function finiteness conjecture

Let $\mathbb{G}$ be a finitely generated group and let $\mu$ be a courteous probability measure on $\mathbb{G}$. Let $\mathsf{LHF}(\mathbb{G},\mu)$ denote the space of Lipschitz $\mu$-harmonic functions. Lipschitz-to-linear finiteness conjecture. If

$$\dim \mathsf{LHF}(\mathbb{G},\mu)<\infty,$$

then

$$\dim \mathsf{HF}_1(\mathbb{G},\mu)<\infty.$$

The authors expect a positive harmonic function arising earlier in the paper to be Lipschitz; this implication would slightly improve their theorem for finitely generated solvable groups and would fit Kleiner's strategy for proving Gromov's theorem.