Advocates say their cause appears to be gaining support among some conservatives, particularly when they can be convinced that a particular bill will not infringe on anyone’s religious rights.

A Senate bill to outlaw discrimination in housing sailed out of committee Monday — with support from a pair of conservative Republicans who had never been on board before.

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Some Republicans have come to see the issue as a matter of personal liberty — and perhaps political survival in an election year when the GOP is struggling to appeal to voters in once-friendly suburbs.

Even so, LGBT rights remain a tough sell in this Southern capital. A bill to repeal the state’s already defunct ban on same-sex marriage failed in a tied committee vote last week, on the opening day of the General Assembly. On Monday, legislation to expand the state’s hate-crime law to include gay and transgender people died in a Senate committee.

The Family Foundation of Virginia, a conservative advocacy group, has lobbied vigorously against such legislation, contending that it threatens religious liberty.

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“When it comes to laws adding the concepts of ‘sexual orientation’ and ‘gender identity,’ we have seen full well by now that these laws have the direct effect of weaponizing governments with the ability to punish those with a viewpoint that runs contrary to its own when it comes to marriage and human sexuality,” the foundation’s president, Victoria Cobb, wrote on the group’s website.

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Robinson considers herself a social moderate. In 2013, she voted for the state’s first openly gay judge. But she has also supported legislation that could allow a private business owner with deeply held religious convictions — akin to the wedding-cake baker who won a narrow victory at the U.S. Supreme Court last year — to turn away gay customers.

She said she intends to stick with her case-by-case evaluation of gay rights legislation even as she advocates for her housing bill.

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“I don’t want people to put me in a box here,” she said. “I have a lot of friends who are in the LGBT community, and they just want to be living their lives. Just like someone who’s heterosexual, they just want to live their lives.”

Robinson drew attention to her bill with an afternoon news conference organized by Equality Virginia, the LGBT rights group that is marking its 30th year. Two polls commissioned by Equality Virginia over the past year found a majority of GOP primary voters in Virginia say anti-LGBT discrimination should be outlawed in housing and public employment — results that line up with public polling on the topic nationwide.

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Appearing with Robinson were two legislators who have supported similar bills in recent years, Sen. Siobhan S. Dunnavant (R-Henrico) and Del. Glenn R. Davis Jr. (R-Virginia Beach).

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“To me, it’s inherently a Republican principle — everybody should have equal opportunity,” Dunnavant said. “We should be a meritocracy. We shouldn’t have any discrimination for any reason.”

For the past three years, the full Senate has approved legislation to protect LGBT Virginians from discrimination in housing and public employment, but the bills have always died in the House.

Davis predicted that the housing measure will have “a good opportunity to get through [the House] this year.”

On Monday, the Senate General Laws and Technology Committee passed separate housing and employment bills, sending them to the floor. Two Republicans signed on to the housing measure for the first time: Sens. Bryce E. Reeves (R-Fredericksburg) and Frank M. Ruff Jr. (R-Mecklenburg). Sen. Richard H. Black (R-Loudoun), one of the legislature’s most outspoken conservatives, cast the only no vote in committee, although Reeves and Ruff joined him in opposing the employment bill.

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James Parrish, executive director of Equality Virginia, said Robinson’s sponsorship of the housing bill in the House could improve its chances for success. The GOP controls both chambers by two votes apiece. “It’s clearly helpful to have a member of the party in power introducing legislation, especially in election years,” he said. “We believe for many reasons, from the work we’ve done to the polling that we’ve seen, to the commitments that we’ve had from delegates on these two issues, that these bills should move forward.”