Conjecture on monotonicity of successive average stack-sorting depths

Let $\mathcal{D}_n'$ denote the auxiliary average stack-sorting depth sequence defined in the paper. Consider its successive differences

$$\mathcal{D}_{n}'-\mathcal{D}'_{n-1},\qquad n\geq 2.$$

Monotonicity conjecture. The sequence

$$\left(\mathcal{D}_{n}'-\mathcal{D}'_{n-1}\right)_{n=2}^{\infty}$$

is monotonically increasing.

This conjecture is motivated by numerical evidence and is presented as a future direction; no proof or resolution is supplied in the given text.