Let us assume the Riemann Hypothesis. Carneiro, Milinovich, and Soundararajan [proved in 2017][1] that for any $x>4$, there exists a prime in $[x,x+\frac{22}{25}\sqrt{x}\log x]$. Using this result, it is straightforward to prove the OP's conjecture for $x\geq 10^{47}$ (under the Riemann Hypothesis).

See also my response to this [related MO question][2].


  [1]: https://arxiv.org/abs/1708.04122
  [2]: https://mathoverflow.net/questions/312236/consecutive-prime-gaps-and-explicit-bound/