Determine the minimum factor for plane E1c extensions

Prove that the smallest possible realization-count multiplication factor for a type E1c 1-extension of a minimally rigid plane graph is $12/7$.

Background

The smallest factor found experimentally for type E1c extensions is approximately 1.71, equal to $12/7$, and this value occurs throughout the complete data for graphs up to 13 vertices. The authors state that these data motivate a conjecture about the global minimum.

References

Since, we have a complete data set for up to 13 vertices, where this lowest factor is the same for all $7 \le n \le 12$ we might conjecture that it is indeed the lowest possible.

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 8, subsection “Plane,” subsection “Extension Constructions”