Perhaps in view of the comments the following clarfication is helpful and since it also sort of an answer and a bit long I give it as answer.
It is quite likely that this conjecture is false yet no counter example was found so far.
The point is, which is mentioned by OP but got a bit blurred via the comments, that Granville argues that large gaps between successive primes in fact will not be bounded by $(\log p_n)^2$ (contrary to what Cramér conjectured) but that one actually needs a constant factor larger than one, and more specifically that there will be infinitely many gaps of size (up to lower order factors) $2 e^{-\gamma} (\log p_n)^2$.`
For details on Granville's arguments, based eg on results by Maier, see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf