Density-property characterization of affine space

Let $X$ be a Stein manifold diffeomorphic to ${\mathbb C}^n$ and having the density property, meaning that the closure of the Lie algebra generated by the complete holomorphic vector fields on $X$ is the Lie algebra of all holomorphic vector fields on $X$. Density-property conjecture. Then $X$ is biholomorphic to ${\mathbb C}^n$. This conjecture asks whether the density property, together with the topology of affine space, characterizes ${\mathbb C}^n$ among Stein manifolds. The supplied text presents it after noting a counterexample to a related question of Diederich and Sibony, but gives no resolution of this statement.