I would be interested to see these studies.
In any case, exercise is, for the most part, when done safely and in a way that allows one to preserve a healthy lifestyle, similar to eating, drinking water, etc. conducive to learning as per the student.
To note, language acquisition s split up into L1 (fluent) and L2 (proficient), where literature does not even mention L3 except in the rarest of cases where learning has occurred outside all critical windows.
It is very easy to see méta-analyses, for which the more studies, the easier it is to find a correlation. Regardless, it can be answered on a simpler level. On a student's needs-basis.
Since Maslow's has been proven faulty time and again, it's important to cater to every individual differently. In specific cases, as was my own growing up, exercise exposed me to allergens, which in my culture, the concept of "allergies" was fictitious and nonsense. I would end up feeling unwell and ultimately needing to sleep after exercising.
Even indoors, severe sensitivity to dust lead to issues as when exercising the blood vessels dilate in order to allow more air to flow through, which afterwards, results in a mysery unlike any other.
Of course, I'm providing a very extreme anecdote for the sole purpose of showing that generalization especially when it appears that later on in the response, it appears there is no grasp of how language is processed nor acquisition occurs in the dichotomy discussed
despite sincere in intent, the writing if taken at face value could lead many astray.
Ultimately, the effects that exercise plays as a general health concept is important, but as an incorporated modality of instruction varies on the individual, and the spacing from when they begin their language acquisition. In order to receive the best results in both didactics and pedagogy (not interchangeable), an organic environment for exchange in the target (language to be learned) should be sought, and the language should be learned « dans le vide » completely separately and independent of their other spoken languages. If they happen to be bilingual or multilingual, this would be another L1, if this is a language learned later in life having missed important critical windows of language development, this would be an L2.
If, the pupil so happens to enjoy exercising, isn't able to sit still, and likes exploring, then it is the job of the individual instructor to keep their student interested, engaged, and attentive to where their student is cognitively.
Of everything one could imagine, the two most vital and core components to language acquisition are 1) that they are making meaningful use of the language, not in the way that it is sensical to others, but rather that it is essential, enjoyable, and significant. The second, I also mentioned that being without constantly referencing another language, because this results in slower language processing that, from an initially very neuroplastic brain becomes hardwired. This extra step in code switching, using the IFJ, requires not unlearning, but enforced learning of another method for which the neuronal connections and myelin sheath need to be so much more reinforced that eventually that initial connection gets pruned.
However due to the strong desire of not necessity of human beings to communicate in a meaningful and heard way, this extremely difficult mental process is often impossible to completely do without.
Getting back to specifically exercise, if it is done, AFTER one has studied, as a general statement, not just with language, this often helps. Especially for individuals who are able to sleep easier after a good exercise (if this is the opposite case for you, exercise before school instead). Sleep is when synaptic pruning or unused areas that have withered are pruned while pathways that have been constantly reinforced develop thicker myelin sheaths.
Due to the fact that children, adolescents, and young adults tend to use their pre-frontal cortex more than adults (shame on us), despite it not having fully matured yet (usually mid 20s), taking one's mind off of the task from overanalyzing and getting lost without help, further going on more tangents, and coming up with one's own ideas without the proper context can often (not always) but often can lead one astray.
So, any activity after studying, an enjoyable one even would be a good association to make and would likely be beneficial for retention. Briefly flipping over material before sleeping, but not studying could help too (literally flipping).
