Concerning lower bounds, we don't even know that the left-hand side is positive for every sufficiently large $X$. The best result of this kind is that, for every sufficiently large $X$,
$$\pi(X+X^{0.525})-\pi(X)\geq\frac{9}{100}\frac{X^{0.525}}{\ln x}.$$
See the last display in [Baker-Harman-Pintz: The difference between consecutive primes, II][1]. Note that $X^{0.525}$ is much larger than $\ln^2 X$, and the above result is the state-of-the-art.

Concerning upper bounds, [Maier (1985)][2] proved the surprising result that there is a constant $c<1$ such that for any $X_0>0$ there exists $X>X_0$ satisfying
$$\pi(X+\ln^2 X)-\pi(X)<c\ln X.$$

  [1]: http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf
  [2]: https://projecteuclid.org/journals/michigan-mathematical-journal/volume-32/issue-2/Primes-in-short-intervals/10.1307/mmj/1029003189.full