Let us assume the Riemann Hypothesis. Carneiro, Milinovich, and Soundararajan proved in 2017 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.