There's no firm ID on the Rex dice, but I've always suspected they are from the Rex Hotel in Chicago. There were lots of places using that name, but these are typical Chicago style dice, made by a Chicago mob owned manufacturer (Taylor & Co).

The Rex Hotel at 2138 S. State, in Chicago was a sleezey saloon with gambling and prostitution upstairs. It was managed by Dennis "Duke" Cooney. In the 1920s Johnny Torrio was a part owner. They say the Chicago police used to stand guard outside.