I think you CAN keep all four right now. Hayward and Favors are up for extensions this summer. Neither is getting the max this summer. If you can get Favors for 4 years 10 per, and Hayward 4 years 12-13 per, then that means you can still sign TWO max players. Burks will NEVER be a max guy, so let's say he gets 10 per. That equals 35 million for three players, not including Kanter. Now, I believe that Kanter WILL be a max guy, so he gets 15 million. That means you now have four players at 50 million. That still leaves you 15 million to fill in the gaps. That is a dang good team.
The question now becomes, what is more valuable? A very, very good pg, or Hayward/Burks?
I don't know the answer to that. Probably depends on who your coach is.
That being said, I bet money that Favors is resigned for around 10, Hayward gets around 12, and Kanter gets the max next summer. I also bet anything that the Jazz try to move Burks for a superstar/really high draft pick.
Now, IF Favors gets resigned for 12+ a year, I completely change my tune on Favors. I would then try to trade him for a superstar. For a guy with no offensive game, 14+ is too much. He can be an elite defender, but he will hurt the Jazz overall at that cost.