The planar 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 $\mathbf{c}_{2}(G)$ denote the planar realization count. The planar E1a doubling conjecture. Then

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

The conjecture formalizes experimental evidence that a planar E1a extension gives exactly the known minimum factor of two. Its general validity is not established in the supplied text.