In the case of the cube, it was shown by Bourgain (very recently) that the constants remain independent of $n$ for all $p>1$. It seems to remain open for more general centrally symmetric convex bodies. On page 3 of his recent preprint, Bourgain writes:
"While it is reasonable to believe that this statement holds in general, our argument is based on a very explicit analysis which does not immediately carry over to other convex symmetric bodies."