Popular science literature is replete with the necessity for superstring theories to live in at least 10 dimensions, requiring at least 6 of them to be compactified so that they are not observable. But can they predict that 4 dimensions are not compactified? Or conversely, for a superstring theory in e.g. 10 dimensions, are there arguments against compactifying 7 dimensions? or only 5? Arguments other than being ruled out by observation I mean.

My question has been marked as a duplicate of Q10651. Only the answer A10723 is relevant to my question. It quotes a paper "Relaxing to Three Dimensions by Karsh and Randall which seems to address my exact question:

We propose a new selection principle for distinguishing among possible vacua that we call the “relaxation principle.” The idea is that the universe will naturally select among possible vacua through its cosmological evolution, and the configuration with the biggest filling fraction is the likeliest. We apply this idea to the question of the number of dimensions of space. We show that under conventional (but higher-dimensional) FRW evolution, a universe filled with equal numbers of branes and antibranes will naturally come to be dominated by 3-branes and 7-branes. We show why this might explain the number of dimensions that are experienced in our visible universe.

But this is far too advanced for me as I can barely recognise buzzwords! And answer A10723 is of rather low quality, just a vague description of the goal of the paper which does not give any hindsight into the physics in there. In any case, this also mean that the other question, Q10075, mine is marked as a duplicate of, has even poorer answers than they look like at first glance. Following the cited by link on the page of Karsh and Randall paper turns out several citations which look to address my question and Q10075, but again at a level I can't grasp.