The spherical E1a doubling conjecture for minimally rigid graphs

Let $G$ be a minimally rigid graph, and let $G'$ be obtained from $G$ by a 1-extension of type E1a. Let $\mathbf{c}^{\circ}_{2}(G)$ denote the spherical realization count. The spherical E1a doubling conjecture. Then

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

The claim extends the observed factor-two behavior to spherical realizations; the supplied text reports it for graphs with at most 13 vertices but does not establish the unrestricted statement.