Embedding open connected Riemann Surfaces in $\mathbb{C}^2$

One of the first example of proper embedding of open Riemann surface is,

Proper embedding of open unit disk in $\mathbb C^2$ which is one of corollary of Fatou-Bieberbach domains (holomorphic dynamics in $\mathbb C^2$). There are results about proper embedding of annulus as well.

But general question is still open which is known as Bell-Narasimman conjecture.

Bell-Narasimman conjecture has two parts (both are open) -

1) Every open Riemann surface can be embedded in $\mathbb C^2$

2) Every embedded open Riemann surface can be properly embedded in $\mathbb C^2$.

This is an open problem, known as

Bell-Narasimhan conjecture. Every open Riemann surface admits a proper holomorphic embedding into $\mathbb{C}^2$.

Look at F. Forstnerič's book Stein Manifolds and Holomorphic Mappings, Problem 9.10.1 p. 446.