"We heard yesterday they are going to be deported today," said Susan Olvera, of Alerta Migratoria NC. "This is imminent deportation. We see this as an emergency."
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A U.S. Immigration and Customs Enforcement (ICE) spokesman said he can't discuss future operations but confirmed the brothers are still in custody in New Mexico.
He said their case has been extensively litigated and that they have no legal basis to remain in the country.
"Yes, he might have come illegally but he does not have a criminal record," said the daughter of Dario Delgado, Yhairi Delgado-Garcia. "He is not a threat to this society. I don't get why they would want to deport him."
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Family members said the brothers have been living in the country illegally for about 20 years. Their children are all U.S. citizens and live in the Triangle.
An Alerta Migratoria NC news release said three children have medical conditions and need monitoring.
"I don't know if we'll be able to financially be stable without him and we need him a lot," said Stephanie Delgado-Garcia, daughter of Jose Delgado.
The children also said their fathers' absence has taken an emotional toll on them.
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"I'm worried and sad because I don't get to see my dad," said Jennifer Delgado-Garcia, daughter of Dario Delgado.
"The reason we are here is to call awareness to our Republicans in North Carolina to let them know how fiscally irresponsible deportation is," Olvera said.
Emily Weeks, the Press Secretary of the North Carolina Republican Party, sent this statement:
"While it's not our place to evaluate this case specifically, the North Carolina Republican Party believes immigrants have undeniably made great contributions to our state and nation, but any national immigration policy must put the interests of our existing citizens first. To start, our border must be secured and illegal immigration must be stopped. Then, and only then, can we begin reforming our system in a way that lets new immigrants experience the American Dream while also benefiting the American economy."