Many immigrants live in areas that are more impoverished and high-density, both of which have been factors historically for the spread of diseases. Additionally, the BCG vaccine—a vaccine known to lead to false positives—is commonly administered in countries outside of the United States, which may possibly contribute to some positive cases but cannot account for all of them.
If one was presented with only these factors, they might draw the conclusion that immigrants from any country administering the BCG vaccine who moved to the United States and live in impoverished high-density areas would be the highest demographic across the board for tuberculosis—and this conclusion would be more-or-less correct, as within each race, immigrants of a certain race tend to have higher rates of tuberculosis than their U.S.-born counterparts. However, if these were the only factors to affect tuberculosis rates, then Asians (specifically Asian immigrants) would not be consistently leading in the diagnosed tuberculosis case rates by so much for so long. Another factor to consider is the higher rates of drug-resistant tuberculosis in Asians, but even still, this doesn't explain the huge disparity.
After investigating factors such as living conditions related to wealth, vaccination history of immigrants, and rates of immigration to high-density areas, it's clear that the answer goes beyond purely socioeconomic factors. If the answer cannot be found in socioeconomic factors, then could it be found in cultural ones? Disease spread and disparity is a multifaceted issue, and without a doubt there are more factors to be considered even once cultural ones are weighed in as well.