Factorial-minus-one bump conjecture for $B(n)$

Define

$$B(n)=h(n!-1),$$

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

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

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

This is presented as the analogue of the factorial-plus-one bump conjecture for $B(n)$; the supplied text gives no proof or resolution.