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

Define

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

with the definition appearing in the paper.

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

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

This is presented as one of the analogous bump conjectures for the four functions associated with factorials; the supplied text gives no proof or resolution.