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The conformal anomaly indicates the breaking of conformal symmetry (angle-preserving transformations) in the quantum theory by quantum fluctuations and is a close cousin of the gravitational anomaly. We show, for the first time, that the conformal anomaly controls the variance of the local magnetization $M_{loc}$ at finite temperatures in spin chains and spin ladders. This effect is perceived at constant and variable temperature across the sample. The change of $M_{loc}$ induced by the conformal anomaly is of the order of 3-5\% of the maximal spin at one Kelvin for DIMPY or CuPzN and increases linearly with temperature. Further, for a temperature gradient of 10\% across the sample, the time-relaxation of the non-equilibrium $M_{loc}$ is of the order of nanoseconds. Thus, we believe that experimental techniques such as neutron scattering, nuclear magnetic resonance~(NMR), spin noise and ultrafast laser pumping should pinpoint the presence of the conformal anomaly. Therefore, we pave the road to detect the conformal anomaly in spin observables of strongly interacting low-dimensional magnets.