I'm guessing that Trump will.
Why? Because he and the Republicans, as the governing party, have more to lose if the government actually shuts down.
And because Trump, in the end game, is pretty good at making a deal, and likes to brag that he can get things done.
And because the Democrats realize they have the cards this time, and are prepared to hang tough.
And because, at different times, Trump has clearly said he wants a deal on the DACA Dreamers, as do lots of Republican members of the House and Senate. That deal was all but done, and is in suspension only because of Trump's foul mouth.
And because Trump wants to turn the page and not let this mess define his leadership.
The outlines of the deal are very simple. Trump gets to keep the government open. He gets part of his enhanced border with Mexico in the form of an actual wall, but not the entire 1,933 miles. He gets tougher standards on visas.
The Democrats get permanent DACA and a path to citizenship for Dreamers. And they get funding for The Children's Health Insurance Program (CHIP).
That's the deal. The deal may happen tonight, or there may be a very brief shutdown before it is consummated.
I could be wrong, of course. With Trump, you never know. But as much as Trump may resent the “deep state,” we can't operate without a government and he is the president.