I wouldn't be surprised if he does but I am not expecting it to happen.
As of today the Brewers would be 25th. The teams that could benefit from this are the Yankees, Marlins, Reds, Giants, Twins, Red Sox, Dbacks and the Cubs. All of these teams have a higher priority than the Cubs. The Cubs are 20th in the order so it is possible one player gets to them but unlikely. The Yankees and Reds need pitching so I don't think Giolito gets past them or past the Giants or Red Sox. Miami could use another bat so Renfroe or Grichuk make some sense for them. The Twins could use a bullpen arm so Moore makes some sense there also. The Giants could also benefit from Giolito as he would fill a rotation hole.
I expect all of the players to get picked up before some of the current playoff leaders even get to pick. I think we can expect a rule change this off season to deal with this just not sure what they are going to do.