This has been studied, but the results tend to argue that it's purely genetic, eg)

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6669050/

> Studies that have measured changes in endogenous lactase activity after an intervention period consistently show a lack of enzyme induction, suggesting that lactose intake does not affect an individual's lactase activity. Although these studies are scarce and have relatively few subjects, data from cross-sectional studies support the theory of purely genetic regulation