= Solution
A random <Pauli Z gate> with equal application and nonapplication probabilities gives
$$
\boxed{\rho'=\tfrac12\rho+\tfrac12Z\rho Z
=\begin{pmatrix}|\alpha|^2&0\\0&|\beta|^2\end{pmatrix}.}
$$
This is exactly the same <dephasing channel> as the unrecorded measurement, for every input <density operator>, including one entangled with an auxiliary system. Individual trials differ: conditioned on the classical random choice, the state is the <pure state> $|\psi\rangle$ or $Z|\psi\rangle$, generally neither a computational <basis vector>. Knowledge of that choice permits recovery by the same <Pauli Z gate>; a measurement outcome does not restore an unknown destroyed superposition.
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