Does there exist a positive integer $k$ such that $(g_i^k)_{i=1}^{\infty},\ g_i^k=p_{i+k}-p_i$ is non-decreasing for all sufficiently large $i$, where $p_i$ is the $ith$ prime?
$g_{i+1}^k\geq g_i^k\iff g_{i+k}^1\geq g_i^1$. A reason to suspect the answer is negative: Assume the answer is positive and $i\geq C$ is sufficiently large. Then $g_{i+tk}^1=2\implies g_{i+(t-1)k}^1=2\implies\cdots\implies g_i^1=2$. In other words, all sufficiently large twin primes would have to lie in arithmetic progressions with common difference $k$.
This seems like a bad approach given the current state of mathematics (and my understanding of it), but I'm optimistic that someone with a background in number theory can provide a better one.
I'm not sure if there's any elementary proof of this fact - though it seems intuitive that it ought to be false, since it's a fairly strong condition on the primes being distributed in some regular fashion. In particular, it states that there can be no run of more than $k$ primes which are "unusually" close together - that is, the primes grow less dense at some regular rate, and the length of any exceptional cases is limited.
However, your conjecture is known to be false. You may likely be aware of the result of Yitang Zhang, who proved that there are infinitely many prime gaps not larger than 70 million. Using a similar method, it has been proven that $\liminf_{n\rightarrow\infty}p_{n+k}-p_n$ is finite, which, of course, implies that the sequence is not non-decreasing (since if it were non-decreasing and bounded, it would mean that, after a point, the primes coincided with an arithmetic progression, which is entirely preposterous).