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In this article we reduce the geometric stability conjecture for the scalartorus rigidity theorem to the conformal case via the Yamabe problem. Then weare able to prove the case where a sequence of Riemannian manifolds isconformal to a uniformly controlled sequence of flat tori and satisfies thegeometric stability conjecture. We are also able to handle the case where asequence of Riemannian manifolds is conformal to a sequence of constantnegative scalar curvature Riemannian manifolds which converge to a flat torusin $C^1$. The full conjecture from the conformal perspective is also discussedas a possible approach to resolving the conjecture.

Almost non-negative scalar curvature on Riemannian manifolds conformal to tori