Connecticut’s Democratic U.S. senators plan to re-introduce a universal background check bill this week that was first proposed six years ago as a response to the massacre at Sandy Hook Elementary school.

Back then — to the dismay of family members of Sandy Hook victims who watched the vote from the Senate gallery — the background check bill failed to win enough votes, even though the chamber was controlled by Democrats at the time.

Republicans currently hold a 53-47 vote majority in the Senate, but most legislation in that chamber requires 60 votes to pass.

At a press conference in Hartford Monday, U.S. Sen. Chris Murphy, D-Conn., was cautiously optimistic about the bill’s prospects.

“I think the stronger the bipartisan vote in the House of Representatives, the more pressure there will on Republicans in the Senate,” Murphy said. “It is not an easy road to get to 60 in the Senate, but it is not impossible and I don’t think you can calculate how much energy will be created by this bill passing the House of Representatives for the first time.”

U.S. Sen. Richard Blumenthal, D-Conn., said public sentiment about gun control has shifted and the National Rifle Association’s grip on members of Congress has slipped.

“We have reached that tipping point when we are breaking the grip of the NRA and the special interests who have been in our way up to now on this issue,” Blumenthal said.

The senators’ bill would extend FBI background checks, which are now conducted when someone purchases a gun from a store, to people buying weapons at gun shows and through the internet.

Murphy said the bill “will make sure that every single person that tries to buy a gun in this country proves that they are not a dangerous criminal before they get their hands on a dangerous weapon.”

“It is simply common sense that you should not be able to get your hands on a deadly weapon if you have a history of violence, if you have a history of serious illness,” Murphy said.

Murphy launched a successful filibuster, holding up work in the Senate, in 2015 to force GOP leaders to schedule votes on the universal background check bill and other gun-related legislation. But, once again, the background check bill failed to garner enough support to pass.

However, leaders in the new Democratic-controlled House have planned votes on more than a dozen gun-related bill, including the universal background check legislation.

Despite the obstacle GOP leadership in the Senate poses to Democrats who seek to tighten federal gun laws, Blumenthal said they are not settling for background checks alone. He also plans to introduce bills that would require safe storage of weapons and extend Connecticut’s “red flag” law to the rest of the nation. That bill would allow family members and others to seek the temporary confiscation of guns from people who could be a threat to themselves or others.

“Background checks are a beginning, they are a bare minimum,” Blumenthal said.