This experiment has actually been done, measuring the precise rate of diminishment of gravity over very short distances. So far as we can tell gravity does follow an inverse square law, and the precision of the experiment means that if it does deviate from inverse square, the extra dimensions can be no bigger than around half the width of a human hair. Any bigger than that and the experiment would have detected the deviation.

https://www.scientificamerican.com/article/string-theorys-ex...