Ahlfors and Beurling (Ahlfors, Lars; Beurling, Arne Conformal invariants and functiontheoretic nullsets. Acta Math. 83, (1950). 101–129.) provided an explicit example of a domain in $\mathbb C$ which allows nonconstant bounded holomorphic functions but allows no injective bounded holomorphic functions. Therefore this domain is not biholomorphic to any bounded domain but allows a "light" mapping to a bounded domain, where "light mapping" means that the preimages of every point are totally disconnected. My question is about such examples in several variables. Do there exist a domain in $\mathbb C^{n}$, $n>1$ which is not biholomorphic to any bounded domain but allows a light mapping to a bounded domain? Clearly a Cartesian product of two copies of AhlforsBeurling domains will allow a "light" mapping to a bounded domain but there is some chance (I don't know) that it will eventually be biholomorphic to a bounded domain. A somehow simpler question is whether on some domain the algebra of bounded holomorphic functions does not separate points but does locally separate them. If such domain exists that would provide an example I request. (I actually doubt the existence of a domain as in the second question).
