conjecture about prime numbers and distance between them

Question

is there a name for this conjecture?

Conjecture: given $p_n$ a prime number sequence where $p_1=2,p_2=3,\cdots$, then for all $n\in\mathbb{N}^*$ and $k\in\mathbb{N}$, holds that $\displaystyle p_{n+2+k}-p_{n+1}\le\sum_{i=0}^{k}p_{n+i}$