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The only two even remotely likely Republican senators to oppose Brett Kavanaugh are Susan Collins (ME) and Lisa Murkowski (AK), and their opposition is less likely than you may think. Both are feeling pressure at home, Collins from pretty much everyone, and Murkowski from the Native Alaskan population that got her elected last time around and is justifiably concerned about the hostility Kavanaugh has demonstrated toward native groups.
Neither is showing a great deal of inclination to oppose Kavanaugh, though. Murkowski is more of a question, since she's playing this very carefully, but Collins has all but showed her hand. She needs a reason to go out on a limb with leadership, and thus far the threat to her 2020 re-election from irate constituents in Maine hasn't been enough of one. Neither, though, has any incentive to buck Donald Trump and Mitch McConnell if their votes are going to be wasted.
And those votes will be wasted if Democrats don't hold together. So far, Senate Minority Leader Chuck Schumer has refused to whip against Kavanaugh. That's despite the fact that Democrats on the Judiciary Committee did some very heavy lifting and real damage to Kavanaugh. With just a fraction of the documents relating to his long career in politics being made available, Judiciary Democrats gave the whole team every possible reason they could need to justifiably oppose this nomination. They demonstrated four ways to Sunday that Kavanaugh is not fit to serve on the highest court in the land, from his repeated instances of perjury to his refusal to commit to not being Trump's get-out-of-jail-free card on the Supreme Court.
Even if there are red-state Democrats who don't want to get into the politics of Trump's corruption, fine. They can oppose Kavanaugh on the basis of the threat he poses to health care. Even Joe Manchin (WV) and Heidi Heitkamp (ND) have been politicking on the threat posed by a lawsuit that would take away protections for people with pre-existing health conditions, which could very well come before the Supreme Court.
There is no reason for Schumer not to be whipping Democrats, because there's no reason for Democrats not to be united in opposition to Kavanaugh. If Democrats are united, all the pressure is on Collins and Murkowski to do what their constituents are demanding. Defeating Kavanaugh—and defeating Trump—is actually within our grasp. Schumer has to make it happen.