It is not know that RH implies EH, or that EH implies RH. Let us denote
$$S(x):=\sum_{p_n < x} (p_n -p_{n-1})^2.$$
Assuming the Lindelöf hypothesis, [Yu (1996)][1] proved that $S(x)\ll_\varepsilon x^{1+\varepsilon}$. Unconditionally, [Stadlmann (2022)][2] proved that $S(x)\ll_\varepsilon x^{1.23+\varepsilon}$.


  [1]: https://doi.org/10.1112/blms/28.3.242
  [2]: https://arxiv.org/abs/2212.10867