The density-property characterization of RR-spaces

Let $X$ be a Stein manifold. It has the density property when the Lie algebra generated by its complete holomorphic vector fields is dense, in the appropriate topology, in the Lie algebra of all holomorphic vector fields on $X$. An $RR$-space is a complex manifold satisfying the Rosay–Rudin-space axioms described in the paper.

Density-property characterization. $X$ satisfies the density property if and only if it is an $RR$-space.

The density property is a central source of holomorphic automorphisms and is known for several important classes of Stein manifolds. The claimed equivalence would characterize $RR$-spaces intrinsically among Stein manifolds, but the paper presents it as open.