Sufficiency conjecture for reflection-symmetric frameworks in quadrilateral normed planes

Let $\|\cdot\|_{\mathcal{P}}$ be a polyhedral norm on $\mathbb{R}^2$ whose unit ball $\mathcal{P}$ is a quadrilateral. Let $G$ be a finite simple graph, let $\theta:{\mathcal{C}}_s\to\operatorname{Aut}(G)$ be an action, and let $\tau:{\mathcal{C}}_s\to\operatorname{GL}(\mathbb{R}^2)$ be a faithful representation such that $s$ swaps the facets of $\mathcal{P}$. Reflection-symmetry sufficiency conjecture. The conditions in Proposition~ are sufficient for the existence of a well-positioned isostatic framework $(G,p)$ in $(\mathbb{R}^2,\|\cdot\|_{\mathcal{P}})$ that is ${\mathcal{C}}_s$-symmetric with respect to $\theta$ and $\tau$. The conjecture was proposed because proving it would require symmetry-adapted Henneberg-type operations on $(2,2)$-tight graphs with a $\mathbb{Z}_2$-action and geometric placement arguments preserving isostaticity; its resolution is not supplied here.