Inoue-surface conjecture for non-elliptic surfaces with multiplicatively automorphic pluriharmonic functions

Let $X$ be a non-elliptic compact complex surface admitting a positive, multiplicatively automorphic, non-constant, pluriharmonic function on a ${\mathbb Z}$-covering. Inoue-surface conjecture. The surface $X$ should be bimeromorphically equivalent to an Inoue surface. A positive answer would complete the classification of compact complex surfaces of Kähler rank $1$ described in the source; the source gives no resolution status.