= Solution
A pure spin-one-half state on the equator of the <Bloch sphere> has $\langle Z_k\rangle=0$, so $|\alpha_k|^2=|\beta_k|^2=1/2$. Its azimuthal phase does not enter the <conditional environment overlap>, because the interaction is diagonal in $Z_k$. Substitution in the <decoherence factor> gives
$$
\boxed{z(t)=\prod_{k=1}^N\cos(2g_kt)}.
$$
The <Zurek spin-bath model> now has a real coherence factor: positive and negative values correspond to opposite relative phases, while suppression of coherence depends on $|z|$.
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