This question was hinted upon with the still open question at [1].
Let $M_n = $ A005250($n$) of the OEIS. That is to say, $M_n = p_{i+1}-p_i$, where $p_i$ is the smallest prime such that $p_{i+1} - p_i > p_{j+1} - p_j$ for all $j < i$.
It was stated at [1] "While the extra information of all of the answers is nice, it appears that proving this conjecture would lead to disprove something with the "heuristic analysis using Cramér's model" and how its is used." Well, this is that question.
What can be proved if the conjecture that $$\frac{M_{n+1}}{M_n} \le 2 \text{ and that}\lim_{n\to \infty}\frac{M_{n+1}}{M_n}= 1$$ is assumed to be true? In particular,does this conjecture prove Shanks in [2], the "Cramér's model"[3], or something else?
[1] Is there a conjecture with maximal prime gaps.
[2] Shanks, Daniel (1964), "On Maximal Gaps between Successive Primes", Mathematics of Computation (American Mathematical Society) 18 (88): 646–651; http://www.ams.org/journals/mcom/1964-18-088/S0025-5718-1964-0167472-8/home.html
[3] http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf
For one the conjecture implies that between any two integers
$p$ and $2p$ where p is a prime belonging to M there is guaranteed to be another prime in M.
Furthermore the right side implies that for any multiplicative constant $h > 1$
There exists infinitely many primes P in M whose density goes up as the magnitudes considered increases such that between
$P$ and $hP$
There exists another prime