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Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\\mathcal{Z}$-stable $\\mathrm{C}^*$-algebras that are either monotracial, or real rank zero with finitely many extremal traces, provided that $\\sigma(x)=\\sigma(y)$ is convex. 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