Let us assume the Riemann Hypothesis. Dudek, Grenié, and Molteni [proved in 2015][1] that for any $x\geq 2$, there exists a prime in $(x-y,x+y)$, where $y:=\frac{1}{2}\sqrt{x}\log x+2\sqrt{x}$. Using this result, it is straightforward to prove the OP's conjecture for $x\geq 10^{40}$ (under the Riemann Hypothesis).

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


  [1]: https://arxiv.org/abs/1503.05403
  [2]: https://mathoverflow.net/questions/312236/consecutive-prime-gaps-and-explicit-bound/