Exact factor for plane E1a extensions

Prove that every 1-extension of type E1a applied to a minimally rigid plane graph multiplies the complex realization count by exactly two.

Background

Type E1a 1-extensions are experimentally observed to have a fixed realization-count multiplier of two. The authors report that the assertion holds for base graphs with at most 12 vertices and formulate the universal statement as a conjecture.

References

We might therefore conjecture \begin{conjecture} Let $G$ be a minimally rigid graph and $G'$ be obtained from $G$ by a 1-extension of type E1a. Then $\frac{2{G'}{2{G}=2$. \end{conjecture}

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 7.1, subsection “Extension Constructions,” immediately before the first conjecture