What is the smallest real $q$ such that there is always a prime between $n^q$ and $(n+1)^q?$

What is the smallest real $q$ such that there is always a prime between
$n^q$ and $(n+1)^q?$

In this answer, it is mentioned that for $q=3$, we are guaranteed the
existence of a prime between $n^q$ and $(n+1)^q$, and that it is
conjectured that this is true for $q=2$. I am wondering though, how close
to $2$ have we gotten? In other words, as of today, what is the smallest
$q$ such that there is always a prime $p$ satisfying
$$n^q <p<(n+1)^q\;\;\text{for all}\; n\,?$$
If possible, I would be delighted to see a link towards a proof of the
answer to this question (or a reference, which I can look up).