Graphical distinct-realisation conjecture from the weak pinned distance conjecture

Let $P$ be a finite set in $\mathbb R^2$, and let $G=(V,E)$ be a finite, simple, connected graph. Write $f_G(P^V)$ for the set of Euclidean distance measurements on the edges of $G$ arising from maps of $V$ into $P$.

Graphical distinct-realisation conjecture. For every $\varepsilon>0$, there exists a positive constant $C_\varepsilon$, independent of $P$, such that

$$\left|f_G\left(P^V\right)\right|\geq C_\varepsilon |P|^{|V|-1-\varepsilon}.$$

The source says that Iosevich and Passant proved that the weak pinned distance conjecture implies this statement. The conjecture itself is not stated as resolved.