The following quote comes from Wikipedia http://en.wikipedia.org/wiki/Prime_gap
"Usually the ratio of $g_n / \log p_n$ is called the ''merit'' of the gap $g_n$;. In 1931, E. Westzynthius proved that prime gaps grow more than logarithmically. That is,
$$\limsup_{n\to\infty}\frac{g_n}{\log p_n}=\infty.$$"
Cramér's conjecture and Firoozbakht's conjecture states
$$\limsup_{n\to\infty}\frac{g_n}{\log^2 p_n} < \infty.$$
Has the folloing been proved?:
$$\limsup_{n\to\infty}\frac{g_n}{\log^3 p_n} < \infty.$$