Change the rule. It smells like a backdoor filibuster to me. And the Democrats will make the whole thing go away if the President withdraws Alito, right? They will then claim one thing had nothing to do with the other. Yeah, right.
("Denny Crane: Gun Control? For Communists. She's a liberal. Can't hunt.")
The Pubs can kill this anytime they want. It only takes a simple majority vote to re-open the session. It takes time to first clear the chamber, then to get everyone back in to vote to re-open.
That's why the Dems say they'll do this every day.