Senate Majority Leader Harry Reid, D-Nev., gave a high-profile boost to Sen. Kirsten Gillibrand on Tuesday by supporting her legislation to remove the prosecution of serious crimes in the military from the chain of command, reports Roll Call.

“I’m going to support Gillibrand, and [Armed Services Chairman Carl] Levin knows that,” Reid told reporters after the Democrats’ weekly caucus luncheon.

But Reid’s backing alone does not appear to be enough to push the New York Democrat’s amendment, which could receive a vote as soon as Wednesday, over the 60-vote threshold. Reid’s position reflects the majority of his rank-and-file Democrats, who prefer Gillibrand’s measure to the Senate Armed Services Committee’s base text. The committee text includes major changes that Gillibrand and her allies say do not go far enough to protect victims of violent crime and sexual assault, or encourage them to report attacks.