I've been trying to understand a little bit better a property that I've noticed on the current list of maximal prime gaps on Wikipedia. It's as follows: $\\$
Problem
Let $g_n$ and $g_{n+k}$ be the $n$-th and $(n+k)$-th prime gaps respectively - both of them also being maximal prime gaps. Does the inequality
$g_{n+k} - g_n ≤ k$
hold for any two maximal prime gaps?
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EDIT:
I removed the restriction that I had earlier that there couldn't be any other maximal prime gaps between $g_n$ and $g_{n+k}$. This is so that the inequality applies to any chosen pair of maximal prime gaps.