Convicts just need water to breed. Cold, cool, warm, dirty, gross or otherwise. They don't care.
My mother keeps Convicts (and a few other smaller Cichlids I've conned her into taking by threatening to use them as feeder fish) She went away on vacation and had her filter fail on her. She came home to another 40 little convicts in the tank and most of her other fish dead. It gets better too, she couldn't find homes for all of them (and wigged when I wanted them for feeders) so she took them down to the flower shop near her with an indoor pond and dumped the rest into the bottom level of the pond (they had just finished it had no fish yet). This was in 2006, as of today, the flower shop still have no idea where all the fish came from, but they, have something like 300 pink cons in the pond.
Live food, clean water and a couple of places to hide seem to be the big thing to getting the South American Cichlids going. Not sure what I've been doing with the Africans, but I'm always finding fry in with those ones.