If the Jazz go under the salary cap, which by all accounts is the plan in order to make the Conley trade work, they are out of money. There are no trade player exceptions for teams under the cap. What some people have said about a Burks exception and more recently a Favors exception is incorrect. As things now stand they can only offer players minimum salaries.
There are only two ways I can see the situation changing. 1) The Jazz could restructure the Conley deal so it involves a third team and doesn't require the Jazz to go under the cap. I think this is unlikely. 2) The Jazz could trade Exum. They could trade him for, say, a pick to a team under the cap, which would give the Jazz additional cap space to work with. This wouldn't surprise me, although the Jazz might have to throw in a pick and/or some cash to get the deal done. Or they could trade him for a player or two who the like and not generate any cap space but keep those players, which I think is slightly less likely but still possible