From Bertrand's postulate we know that when $n\ge2$ there is always a prime between $n$ and $2n$. From the answer here, we know that for $n\ge31$ there is always a prime between $n$ and $\frac65n$. Let $r>1$. I fully expect the following conjecture to be unproven:

There is always an $n_0$ such that when $n \ge n_0$ there is always a prime between $n$ and $rn$.

What is the smallest known $r$ for which this is known to be true?