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Quality observational research is probably very difficult to do since you can't properly control for indirect cognitive benefits you get from learning a second language and I'd take any results with a grain of salt. You also can't properly control for confounding factors e.g. reasons for learning a second language. I think you'd need experimental research with randomization to several languages and this would be very costly and possibly inethical to set up.
I have without a question gotten a huge boost from learning English since there aren't enough texts in my native language about psychology, cognitive science and medicine that happen to be my main interests. My native language also lacks the vocabulary to deal with those subjects efficiently. I have also learned several memory techniques and done cognitive tests and training solely because of being fluent in English.
You just need to have an area where different schools have different curriculums and there a lottery mechanism for deciding which student goes to which school.