For Haar-random state-tangent frames, the fraction of quantum Fisher information kept by a fixed-basis record and then by a rank-r diagonal readout are independent Beta variables, so low-weight readouts retain exponentially little information.
Zamir, A proof of the Fisher information inequality via a data processing argument, IEEE Trans
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Readout-Rank Laws for Isotropic Quantum Tangents
For Haar-random state-tangent frames, the fraction of quantum Fisher information kept by a fixed-basis record and then by a rank-r diagonal readout are independent Beta variables, so low-weight readouts retain exponentially little information.