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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. Note that $X^{0.525}$ is much larger than $\ln^2 X$, and the above result is the state-of-the-art.