Monotonicity conjecture for the number of subset sizes in the Z-curve

Let $D$ and $K$ be parameters of the set $\mathbf{X}^D_K$, and let $n$ be such that $K=n+1$. A subset size is a cardinality attained by a subset of $\mathbf{X}^D_K$.

Monotonicity conjecture. For given values of $D$ and $K=n+1$, the minimum number of subsets of different sizes in $\mathbf{X}^D_K$ is at least equal to the number of subsets of different sizes when $K=n$.

The conjecture concerns how the possible subset-size counts change under one further subdivision of the $Z$-curve; the supplied text gives observational motivation but no proof or resolution.