Dot-product realisation conjecture for locally completable graphs

Let $G=(V,E)$ be a semisimple graph with $|V|\geq4$ that is locally completable in $\mathbb R^3$. Let $g$ be the dot-product measurement map in dimension $3$, and let $P\subset\mathbb R^3$ be finite.

Dot-product realisation conjecture. One has

$$\left|f_{g,G}\left(P^V\right)\right|\gg |P|^{|V|-1},$$

and this bound is tight.

The source says that a similar approach might provide a full solution, so the statement remains prospective and open.