When it comes to unnerving ethics allegations, the U.S. Supreme Court struggled through a difficult spring. Justice Clarence Thomas, of course, was caught up in a multifaceted ethics mess that even his most sycophantic allies struggled to explain away. Soon after, Justice Neil Gorsuch faced a separate ethics controversy of his own. on Wednesday, Justice Samuel Alito joined the ignominious club.

An obvious question soon followed: Can anything be done about the high court, its members, and its alleged ethical lapses?

For his part, Supreme Court Chief Justice John Roberts has indicated that he doesn’t see the need for any kind of judicial reforms or changes to ethics laws, because there’s no point in fixing a problem that doesn’t exist. As a recent Associated Press report summarized, Roberts and his colleagues are of the opinion that when it comes to ethics, “they will set their own rules and police themselves.”

As The Washington Post reported, some on Capitol Hill prefer a more proactive approach.

Senate Democrats said the revelation of the [previously undisclosed Alito] trip, by the news organization ProPublica, was one more reason they would move forward on legislation to tighten ethics rules for the justices. Although there appears to be little interest in the Republican-led House in forcing changes upon the high court, Senate Judiciary Committee Chairman Richard J. Durbin (D-Ill.) said his panel would consider legislation after the Senate returns from its Fourth of July recess.

“The highest court in the land should not have the lowest ethical standards. But for too long that has been the case with the United States Supreme Court. That needs to change,” Durbin said in a joint statement with Democratic Sen. Sheldon Whitehouse of Rhode Island, who leads the Senate Judiciary Committee’s panel with jurisdiction over the federal judiciary.

Given the circumstances, it’s tempting to think even Senate Republicans might hesitate before shrugging with indifference to the high court’s ethics troubles. It’s in everyone’s interest for the public to have confidence in the Supreme Court as an institution, and as the court’s public standing deteriorates, it stands to reason that lawmakers would take obvious, nonpartisan and nonideological steps to help put things right.

But the vast majority of GOP officials just don’t seem to care. Several Senate Republicans defended Alito on Wednesday, and Senate Minority Leader Mitch McConnell went so far as to argue that Congress — which has oversight authority over the judiciary — should simply “stay out” of the Supreme Court’s business.

There are a few pending pieces of legislation intended to address the justices’ ethics standards, including the Supreme Court Code of Conduct Act and the Supreme Court Ethics, Recusal, and Transparency Act. Across the House and Senate, the measures have 103 co-sponsors — but only one Republican. (Alaska Sen. Lisa Murkowski backed the Supreme Court Code of Conduct Act in April.)

 In other words, the odds of a judicial ethics bill overcoming a GOP filibuster in the Senate are effectively zero, and the odds of success in the Republican-led House are probably worse.