To stop unanimous consent all it takes is one senator to voice an objection, which is why this will never happen.
But their audacity to make the claim in the first place is troubling. They'll keep on keepin' on.
That's true, but they can ask for unanimous consent at 0 Dark Thirty, as they did when they saw to it that the Assault Weapons Ban not be repealed back in '95 when the ban was only a year old, and Republicans had gained control of Congress in part because they said they'd repeal the ban.
Similarly the Brady Bill passed because of an agreement, at Oh Dark Thirty, with 3 senators present, to end the filibuster.
The assault weapons ban was not passed as a stand alone bill with a "pure" up and down vote, instead it was inserted into the Crime Bill, which was difficult to vote against in the first place, and easy to justify to your constituents, "even though" it contained the AWB.