We study the deformation of spherical conical metrics with at least some of the cone angles larger than 2π. We show in this note via synthetic geometry that for one family of such metrics, there is local rigidity in the choice of cone positions if angles are fixed. This gives evidence of the analytic obstruction considered in recent works of Mazzeo and the author [20, 21].

The author would like to thank Rafe Mazzeo and Bin Xu for many helpful discussions and suggestions, Alexandre Eremenko for giving insightful comments. The author is also grateful for the anonymous referee for comments. The author is partially supported by the NSF grant DMS-1905398.