L. Ni's existence conjecture for polynomial-growth harmonic functions

Let $(M^n,g)$ be a complete Riemannian manifold of positive sectional curvature. A manifold has maximum volume growth when its volume growth is maximal in the relevant comparison sense, and a harmonic function has polynomial growth when its growth is bounded by a polynomial in the distance. L. Ni's conjecture. The necessary and sufficient condition for $M^n$ to admit nonconstant harmonic functions of polynomial growth is that $M^n$ has maximum volume growth.

The conjecture addresses when positively curved complete manifolds possess nonconstant harmonic functions with controlled growth. The surrounding discussion identifies this as an open question in the source; no resolution is supplied there.