Generalized finite-dimensionality conjecture for $(v,p)$-polynomial-growth harmonic functions

Let $M$ be a Riemannian manifold with non-negative Ricci curvature, let $x_0\in M$ be a fixed point, and write $r(x)=d(x,x_0)$. Let $v$ be a real function. A harmonic function has $(v,p)$-polynomial growth when it has polynomial growth of degree $p$ with respect to $v$, namely when $|f(x)|=O(r^p v(r))$. Generalized finite-dimensionality conjecture. For a fixed integer $p$ and real function $v$, the space of harmonic functions with $(v,p)$-polynomial growth on $M$ is finite dimensional. This is proposed as a generalization of the polynomial-growth harmonic-function conjecture; the source does not state whether it has been proved or disproved.