I have been recently reading quite a little about Kaluza Klein theories. I am still far from mastering this but I am curious if any experiment that may disprove or give hints of the existence of extra spatial dimensions is to be expected to be performed relatively soon.

[This page of CERN][1], for example, says that the LHC might give some clues, but it doesn't really dive into much detail.

So, in some detail, which experiments might say something about this, and when might they be expected? [1]: http://home.web.cern.ch/about/physics/extra-dimensions-gravitons-and-tiny-black-holes

I have been recently reading quite a little about Kaluza Klein theories. I am still far from mastering this but I am curious if any experiment that may disprove or give hints of the existence of extra spatial dimensions is to be expected to be performed relatively soon.

This page of CERN, for example, says that the LHC might give some clues, but it doesn't really dive into much detail.

So, in some detail, which experiments might say something about this, and when might they be expected?