Let $ R(n)$ be the $ n$ th record gap between primes; $ R(n)=\mbox{A005250}(n)$ in OEIS.
The paper arXiv:1709.05508 conjectures, among other things, that $ $ R(n) = O(n^2) \tag{1} $ $ and, more specifically, $ $ R(n) \le n^2. \tag{2} $ $
(The heuristic reasoning in arXiv:1709.05508 is based on (a) Cramer’s conjecture and (b) a conjecture that record prime gaps occur more often than records in an i.i.d. random sequence of comparable length.) From computations, we know that the first 77 record (maximal) prime gaps (between primes below $ 10^{19}$ ) do satisfy $ (2)$ .
Is it possible/likely that $ (1)$ and $ (2)$ are true by accident (and therefore not amenable to proof)?
How likely is it that we might find or prove the existence of a counterexample to $ (2)$ ?
