Minimum factor for planar E1c extensions

Prove that the minimum possible realization-count ratio for a planar 1-extension of type E1c is $12/7$ (approximately 1.71).

Background

Type E1c planar 1-extensions can increase the number of complex realizations by less than two. Exhaustive data through 13 vertices found a lowest factor of approximately 1.71, corresponding to $12/7$, consistently across the relevant vertex counts.

The authors indicate that this observed minimum may be the universal minimum, but do not prove it.

References

Since, we have a complete data set for up to 13 vertices, where this lowest factor is the same for all $7 \leq n \leq 12$ 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 8.1.1, Extension Constructions, paragraph following Table Laman_Increase_H2c