Conjecture on planar 2-edge-apex graphs and flexible rigidity circuits

Let $G$ be a planar graph, and let $e$ and $f$ be edges added to $G$. A graph is $(3,6)$-tight if it has $3|V|-6$ edges and every subgraph $H$ with at least three vertices satisfies $|E(H)|\leq 3|V(H)|-6$; an $\mathcal{R}_3$-circuit is a circuit in the three-dimensional rigidity matroid, and it is flexible or rigid according as its generic rigidity rank is deficient or full. The 2-sum of two graphs is their matroidal 2-sum along a common edge. The following hold. (i) If $G$ is planar and $G+\{e,f\}$ is a $(3,6)$-tight flexible $\mathcal{R}_3$-circuit, then $G$ is the 2-sum of two rigid $\mathcal{R}_3$-circuits. (ii) If $G$ is planar and $G+\{e,f\}$ is a flexible $\mathcal{R}_3$-circuit, then $G+\{e,f\}$ is $(3,6)$-tight. This conjecture is intended to characterize which 2-edge-apex $(3,6)$-sparse graphs are $\mathcal{R}_3$-independent; the provided text gives no resolution of either assertion.