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 $(2)$.

Can we reasonably say that $(1)$ and $(2)$ are true by accident (and therefore not amenable to proof)?

**Edit:** Many thanks @Gerhard and @reuns for your insights!
Based on more thinking, I am now sure that *in Cramer's probabilistic model of primes* we can prove:
with probability 1, the true order of magnitude of the $n$th record prime gap is $O(n^2)$ (I haven't found a reference yet, though).
So, Gerhard's assessment below would mean that the true order of $n$th record gap between ordinary primes is less than that for Cramer's probabilistic primes.