Strict realizability conjecture for $K_{2,3}^{'}$-free graphs

Let $G$ be a graph, let $Y_G$ denote its strict realization space, and let $\mathcal{R}_\ell$ denote the relevant matroid or independence structure. A graph is $K_{2,3}^{'}$-free if it contains no subgraph isomorphic to $K_{2,3}^{'}$.

Strict realizability conjecture. If $G$ is a $K_{2,3}^{'}$-free graph which is independent in $\mathcal{R}_\ell$, then

$$Y_G \neq \emptyset.$$

This gives a sufficient condition for strict realizability. The source presents it as a conjecture following examples of graphs that are not strictly realizable.