The Stein-manifold conjecture for positive constant Bergman curvature

Let $M$ be a Stein manifold. Let $A^2(M)$ be its Bergman space, assumed to be base-point free, meaning that its sections do not vanish simultaneously at any point, and to separate holomorphic directions. The Stein-manifold conjecture. The Bergman metric of $M$ cannot have positive constant holomorphic sectional curvature. The source presents this as an open conjecture, motivated by the question of replacing an infinite-dimensionality assumption in a preceding theorem by Steinness; the paper does not establish it in general.