If Cramér's is proved?

Question

Harald Cramér proved that under this assumption that the Riemann hypothesis is true., the gap $g_n$ satisfies $$g_n = O(\sqrt{p_n} \ln p_n) ,$$

using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that $$g_n = O\left((\ln p_n)^2\right).$$

Would the Riemann hypothesis be true if the $(\ln p_n)^2$ conjecture was proven true? (I know the proof will not happens soon. But, if it does?)

Answer 1

I do not believe that a proof of the Cramér conjecture would imply RH.