A few days ago a question was asked about is there always at least one prime in closed intervals of the form $[2^n,2^n+n^2]$ (endpoints included) and, current state-of-the-art/science, is that we still do not know..

I was thinking a little about more general problem, that is, about primes in intervals of the form $[a^n,a^n+n^2]$ where $a\geq2$ is an integer.

I am supposing, with very little of knowledge about this field, that also this more general problem is open for every $a\geq2$.

But also, it somehow seems irrelevant to me what $a$ we will choose, that is, that "almost always" there will be a prime in such intervals, where, of course, "almost always" means that density of such intervals that contain no primes equals zero. (There is no need at the present moment to go into different possible (equivalent or non-equivalent) definitions of densities)

But even if we do not know is that true for every $a \geq 2$, what can be told about occurences of primes "on average" for different $a$? Should the density be equal for every $a$ we choose(even if it is not equal to $1$, but it seems unavoidable that it is)? Is it equal for every $a$?