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 terms)  $2 e^{-\gamma} (\log p_n)^2$.`

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