Extra dimensions are always a desperate measure. If you add them to your theory you open up a huge parameter space, because you have no clue about the number, size and topology of the extra dimensions. What is missing is a real physical symmetry or reason that would enforce the existence of extra dimensions and on the other side restricts its geometry and topology.

Just adding more parameters to your theory will allow you to overfit the data better, but that does not mean you understand more about nature.