The E1s63 doubling conjecture for minimally 3-rigid graphs

Let $G$ be a minimally 3-rigid graph, and let $G'$ be obtained from $G$ by a 1-extension of type E1s63. Let $\mathbf{c}_{3}(G)$ denote the three-dimensional realization count. The E1s63 doubling conjecture. Then

$$\frac{\mathbf{c}_{3}(G')}{\mathbf{c}_{3}(G)}=2.$$

The conjecture is motivated by experiments in which E1s63 extensions always doubled the number of realizations. The supplied text does not establish the claim for all minimally 3-rigid graphs.