The representation-to-TD-Delaunay conjecture

Let $H_d\simeq \mathbb{R}^{d-1}$ be the hyperplane used for the $d$-dimensional construction, let $R$ be a $d$-representation, and let $\Sigma(R)$ be the abstract simplicial complex associated with $R$. Representation-to-TD-Delaunay conjecture. For every $d$-representation $R$, the abstract simplicial complex $\Sigma(R)$ is a TD-Delaunay complex of $H_d\simeq \mathbb{R}^{d-1}$. The reciprocal statement is known for $d=2$ and $d=3$, while the assertion for arbitrary $d$ is presented as an open problem.