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.


  [1]: http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf