No, the House has some conservative dems, blue dogs, who will hold the key to the outcome. The House has been more conservative on this issue. In Dec 2005, when HR 4437 [enforcement only] passed, there were 36 Dems who voted with the Reps.
That still looks pretty hopeless, if you ask me. We could find only 36 Dems to vote to enforce the law? That pretty much means that all of the GOPers must vote correctly, which isn’t going to happen.