Determine the existence of nonconstant ergodic harmonic functions on higher-dimensional lattices

Determine whether nontrivial ergodic harmonic functions on $\mathbb{Z}^d$ exist for dimensions $d\ge 3$.

Background

The paper explains that, for the real-valued Gaussian free field in dimensions $d\ge 3$, the existence of an ergodic Gibbs measure with infinite specific free energy is equivalent to the existence of a nontrivial ergodic harmonic function on $\mathbb{Z}^d$.

This equivalence links an unresolved Gibbs-measure question to a problem in discrete potential theory. The authors note that such harmonic functions do not exist in two dimensions, but that their existence in dimensions at least three remains unresolved.