Hartogs-type extension for tube-like domains in C²

In this paper we consider the Hartogs-type extension problem for unbounded domains in C². An easy necessary condition for a domain to be of Hartogs-type is that there is no a closed (in C²) complex variety of codimension one in the domain which is given by a holomorphic function smooth up to the boundary. The question is, how far this necessary condition is from the sufficient one? To show how complicated this question is, we give a class of tube-like domains which contain a complex line in the boundary which are either of Hartogs-type or not, depending on how the complex line is positioned with respect to the domain.