Determine the minimum factor for spherical E1c extensions

Prove that the smallest possible realization-count multiplication factor for a type E1c 1-extension of a minimally rigid graph on the sphere is $3/2$.

Background

The minimum factor observed in the spherical computations is $1.5$, attained for every tested graph size from 6 through 12 vertices. The authors explicitly state that this evidence motivates a conjecture that $3/2$ is the universal minimum.

References

Again since we have complete data for graph with less than 13 vertices and since $1.5$ is the lowest factor for all graphs with $6 \le n \le 12$ we may conjecture that $k_1=1.5$.

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