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

Define

$$A(n)=h(n!+1),$$

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

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

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

The paper notes that prime-value heuristics for $n!+1$ support this conjecture, while no proof of infinitely many such bumps is known.