Race and socioeconomic status are pretty strongly correlated but I'd imagine it's possible to do a study to see what the extent of each's influence is. You'd need to find "high socioeconomic" names that are also strongly correlated with race(s) themselves correlated with low socioeconomic status and vice versa which honestly might be the hardest part. The disambiguation from a statistical standpoint doesn't seem that difficult once you have the data.

Another hindrance to collecting data in the first place is that idiots will read this and downvote it because they think statistics are racist, I guess?