Orthotope sum conjecture for Z-curve subsets

Let $\mathbb{T}\subset\mathbf{X}^D_K$ be a subset, and let $\mathcal{O}\subset\mathbb{T}$ be an $D$-dimensional orthotope with $2^D$ elements. Write $S(\mathcal{O})$ for the sum of the elements of $\mathcal{O}$, and let $\Psi$ be the constant associated with the ambient set.

Orthotope sum conjecture. For every orthotope $\mathcal{O}\subset\mathbb{T}$,

$$S(\mathcal{O})=\Psi.$$

The source supports the claim with an explicit decomposition of a $48$-element subset into six orthotopes in dimension $3$, each having sum $\Psi^3_5=131068$, but supplies no general proof or resolution.