Need Clarification on Stability of Matching in Gale-Shapley Algorithm with Uniform Preference Lists

Question

I am working on implementing the Gale-Shapley algorithm for a stable matching problem, where all candidates on one side (let's call them c1 to cn) have the same preference list, and candidates on the other side (x1 to xn) also have identical preferences. Specifically, the preference lists are structured as {x1, x2, ..., xn} for all candidates on one side, and {c1, c2, ..., cn} for candidates on the other side.

I understand that the Gale-Shapley algorithm is designed to produce stable matchings by iteratively proposing and accepting or rejecting proposals based on the preferences of the candidates. However, I want to confirm whether the matching generated in this scenario will indeed be stable.

My questions are:

Will the Gale-Shapley algorithm guarantee a stable matching when all candidates on both sides have the same preference lists? Are there any special considerations or modifications required in the algorithm implementation to ensure stability in this uniform preference list scenario? Are there any potential issues or edge cases I should be aware of when implementing the algorithm with uniform preference lists?