Conjecture on global rigidity of 2-edge-apex graphs

Let $G=(V,E)$ be a graph. A framework is globally $\mathcal{R}_3$-rigid if every framework in $\mathbb{R}^3$ with the same edge lengths differs from it by a composition of isometries. The graph $G$ is 2-edge-apex if deleting two edges makes it planar. Then $G$ is globally $\mathcal{R}_3$-rigid if and only if $G$ is 4-connected and

$$G-e\text{ is }\mathcal{R}_3\text{-rigid for all edges }e\in E.$$

This would extend the preceding characterization from edge-apex graphs to 2-edge-apex graphs. The supplied text provides no proof or resolution of the assertion.