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(CNN) —  

The Department of Housing and Urban Development is proposing a new rule that would allow federally funded homeless shelters to consider sex and gender identity when deciding whether to accommodate someone.

The proposal on Wednesday came a day after Secretary Ben Carson told Congress he is “not currently anticipating” changes to the Equal Access Rule, an Obama-era rule that required shelters provide lodging regardless of gender identity.

The proposed rule would permit shelters that have facilities, like bathrooms and sleeping quarters, separated by sex to establish policy that considers a person’s sex for allowing accommodation or admission to the facility or portion, which critics say would lead to transgender individuals being turned away or discouraged from seeking shelter and lead to a rise in homelessness and discrimination.

The rule would let shelters use “privacy, safety, practical concerns, religious beliefs” as a basis for their decision.

In its proposal summary, HUD said the rule would continue the agency’s “policy of ensuring that its programs are open to all eligible individuals and families regardless of sexual orientation or gender identity.”

During a House Financial Services hearing Tuesday, Carson said HUD’s “responsibility is to make sure everybody is treated fairly.”

Asked if HUD has current or future plans to eliminate the Equal Access Rule, Carson replied, “I’m not currently anticipating changing the rule.”

In 2017, HUD had removed from its website links to training materials to prevent LGBT discrimination in homeless shelters.

Carson argued on Tuesday that the guidance was not necessary, because the 2012 and 2016 Equal Access Rules “adequately provide for fairness for all communities.”

Rep. Jennifer Wexton, D-Virginia, accused Carson of being untruthful or ignorant of his agency’s policy objectives.

“One day after (Carson) told me he isn’t anticipating any changes to protections for LGBTQ people in shelters, HUD announced a proposal to gut that very rule,” Wexton said Wednesday. “He either lied to Congress or has no idea what policies his agency is pursuing. Either way, it’s unacceptable.”

Wexton introduced a bill Thursday that would block HUD from implementing its proposal rule.

She later called on Carson to resign, saying in a statement that he has “proven himself to be deceitful and inept” and “unfit to serve” as secretary.

Wexton claimed that Carson “lied again” when he called her the following day “to ‘clarify’ his testimony.”

Carson said in a statement that he thought Wexton was asking whether HUD was going to remove the anti-discrimination protections for members of the LGBTQ community from the 2012 Equal Access Rule.

“We are not, as I stated,” Carson said. “I later realized she was asking a very technical question about the code of federal regulations on self-identified gender.”

Carson said he called Wexton to “clarify that our intention is to stop treating sex and self-identified gender as the same, because I believe Washington shouldn’t be telling the rest of America how to determine whether someone is a man or a woman.”

Mara Keisling, the executive director of the National Center for Transgender Equality, called the HUD proposal a “heartless attack on some of the most vulnerable people in our society.”

“The programs impacted by this rule are life-saving for transgender people, particularly youth rejected by their families, and a lack of stable housing fuels the violence and abuse that takes the lives of many transgender people of color across the country,” Keisling said in a statement. “Secretary Carson’s actions are contrary to the mission of his Department and yet another example of tragic cruelty of this administration.”

Julian Castro, the former HUD Secretary who helped finalize the 2016 Equal Access Rule, argued on Twitter that “rescinding this rule is a shameful decision that will result in trans shelter-seekers being forced on the streets.”

The HUD rule comes as the Trump administration has banned transgender recruits from joining the military and announced a rule aimed at protecting health care workers who cite moral or religious reasons from paying for or participating in certain services, but critics say would allow LGBTQ individuals to be denied care.