Of course it is still important. I have always hated the term, "open-relationship." IF there isn't a common trust, commitment, honesty with one another, then I really don't see the relationship and bond being a strong one. I've seen too many people give their significant others, "get out of jail free" cards, for cheating, lies, what have you. There isn't a "get out of jail free" card with me. If you are willing to go there with someone that isn't your partner, husband, wife, what have you... then the commitment isn't really there either. If you are willing, then that bond has broken to some extent. It doesn't mean that you can't go back and mend it, but usually people just don't want to at that point. It just seems like, once that part is damaged in a relationship, then everything else: the trust, the honesty, the bond is too.
A relationship's strength is that commitment to the other person. As soon as that begins to falter you better begin to mend it or end it there. Don't go out of your way to find what's lacking somewhere else. Don't go looking for happiness in the colder and tougher times. Once you decide to make a commitment to someone else, close the other doors that are open to you and keep them closed.