Inclusion of distance-reducing move sets

For a matrix $A$, let $\mathcal D(A)$ and $\mathcal D^s(A)$ denote the two distance-reducing move sets defined in the paper.

Distance-reducing inclusion conjecture. For every matrix $A$,

$$\mathcal D(A)\subseteq\mathcal D^s(A).$$

The source presents this inclusion as an open conjecture concerning the relationship between the two distance-reducing move sets.