Existence of non-compact, non-homogeneous harmonic manifolds with exponential volume growth in dimensions ≥ 6

Determine whether there exist non-compact, non-homogeneous harmonic manifolds of dimension at least six whose volume grows exponentially with radius (i.e., exhibits exponential volume growth).

Background

The paper reviews known results indicating that harmonic manifolds with polynomial or subexponential volume growth are flat, and that homogeneous harmonic manifolds are classified as either flat Euclidean spaces, rank-one symmetric spaces, or Damek-Ricci spaces. Within this context, the authors note that the existence of non-compact, non-homogeneous harmonic manifolds exhibiting exponential volume growth in higher dimensions is unresolved.

This question sits at the intersection of rigidity phenomena in harmonic manifolds and the broader classification program distinguishing homogeneous from non-homogeneous examples. Establishing existence (or nonexistence) would have implications for the landscape of harmonic manifolds beyond the known homogeneous cases.