The prime-index inequality conjecture

Let $p_n$ denote the $n$-th prime. Prime-index inequality conjecture. For all integers $n\geq 2$ and all integers $k$ satisfying $1\leq k\leq n-1$,

$$p_n\geq p_k+p_{n-k-1}.$$

The inequality would imply that the function defined by $f(0)=f(1)=1$ and $f(n)=p_{n-1}!$ for $n\geq2$ is an abstract factorial, with consequences for irrationality criteria involving abstract factorials. The source gives no resolution.