Motivation to under-report is certainly up in the last decades with bid conditions and owners and general contractors rewarding good records with job awards and penalizing poor reporting companies with removal. So you have “bid lusts.” No substantive study or sampling has been done on this issue.
OSHA records have historically been flawed by factors of 2 and 3x compared to workers’ comp records and even coroner reports. Of major concern in "unskilled" construction trades all the way up to drywall, masonry , earthwork. Illegal or sub-legal immigrants have a culture to not report, self treat, return to homeland or change employment before doing anything to allow themselves to be exposed to any institutionalized or quasi governmental system - including care center or hospitals or doctors.
Further, that culture does not, on first generation, typically allow cumulative trauma or muscular, ergo-type injuries to be something to admit. And as an unscientific but highly experienced and observant quasi-historian, I'll say this: declarations of the greater Hispanic culture having less of those type injuries is fact. Biomechanics and fitness for task and work hardened people would as a rule be less prone, notwithstanding cultural and other built in biases. This is a good place for some non-political research. The recent OSHA announcement that Hispanics were getting hurt more was a lie - and they blamed management and abuse! For political expediency and a selfish agenda. Facts later proved as a percentage of workforce the RATE of injury was much lower compared to percentages.