Monodromy conjecture for compact LCK manifolds with automorphic potential

Let $M$ be a compact locally conformally Kähler (LCK) manifold with automorphic potential, meaning that $M$ admits a Kähler covering with an automorphic potential. Monodromy conjecture. The monodromy of the weight bundle of $M$ is isomorphic to $\mathbb Z$. The authors explain that a deformation can produce monodromy $\mathbb Z$, but conjecture that this deformation is unnecessary.