Transformation sum conjecture for Z-curve subsets

Let $\mathbf{X}^D_S$ and $\mathbf{X}^D_B$ be the sets at levels $S$ and $B$, let $\mathbb{T}\subset\mathbf{X}^D_S$ be a valid subset, and let $\phi^D_{(S,B)}$ be the transformation between these levels. Write $S(\mathbb{T})$ for the sum of the elements of a subset and let $\Psi^D_B$ denote the corresponding constant at level $B$.

Transformation sum conjecture. For any valid subset $\mathbb{T}\subset\mathbf{X}^D_S$,

$$S\bigl(\phi^D_{(S,B)}(\mathbb{T})\bigr)=\Psi^D_B\times\frac{|\mathbb{T}|}{2^D}.$$

The source reports verification of related behavior in dimension $2$ up to $K=8$, while noting that the transformed subset need not remain valid in dimension $3$; the asserted sum identity is nevertheless left unresolved.