The argument that illegal immigrants in the United States will not self deport is often made to explain why they should just be given amnesty.  As the case with South Korea shows illegal immigrants there self deported once they had a viable option to seek legal status:

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An amnesty program for overstaying foreigners is having an unexpected consequence: Korea is suddenly short of nannies, restaurant help and construction workers.

In April, the Ministry of Justice announced an amnesty for people who overstayed their visas or illegally immigrated. It said that they could apply for overseas Korean visas, known as F-4, if they left Korea and went home in the next six months.

This was the first time the ministry has promised to allow all overstayers to apply for visas to return to Korea.

The amnesty is working. Over 12,000 foreigners who were in Korea illegally left the country from April to May, twice the 5,300 who left in the same period last year, according to the Ministry of Justice.   (…………)

“When I went to Incheon airport at 5 a.m., the third floor was crowded with illegal immigrants waiting to leave,” said Lee Byung-chun, who works for the Han-a administrative agency.    [Joong Ang Ilbo]

You can read the rest at the link, but the big different though with South Korea compared to the US is that there isn’t an entire political party trying to lock in the illegal immigrant vote.  This is likely why Korea was able to successfully implement immigration reform and the United States cannot.