Factorial-plus-one bump conjecture for $C(n)$

Define

$$C(n)=H(n!+1),$$

where $H$ is the function used in the paper.

The $C(n)$ bump conjecture. There are arbitrarily large $n$ such that

$$C(n)<C(n+1).$$

The paper presents this as an analogue of the conjecture for $A(n)$ and notes heuristic support from prime and composite values of $n!+1$; it gives no resolution.