The slaves probably would not have remained slaves, because in the Southern cities, agitation was growing due to immigrant laborers who resented the fact that they had to compete with slave labor for work. Political pressure from the cities (particularly the seaports) would have forced slavery to end by the 1870s, had the South won.
I have no doubt slavery would have ended eventually, but I seriously doubt it would have been a result of pressure from imigrants. Most likely by the 80's or 90's pressure would have come from foreign countries who consumed the South's cotton. The South learned a hard lesson when they discovered that Egypt easily picked up the slack for the world's loss of the South's cotton.
That's ridiculous. Look at the census data for 1870. In most southern states the percentage of foreign born people were 1 or 2 percent of the population. There was no immigrant population competing with slave labor for work because there was virtually no immigrant population to begin with. And what population there was wasn't competing for work commonly held by blacks.