= Solution
Write $H=gI_S\otimes Z_D\otimes X_E$, with $Z_D=\Lambda_\uparrow-\Lambda_\downarrow$. The orthogonal <projection operators> obey $\Lambda_\uparrow\Lambda_\downarrow=0$ and $\Lambda_j^2=\Lambda_j$. Hence $[Z_D,\Lambda_j]=0$, and the remaining tensor factors commute with detector operators. Therefore
$$
\boxed{[H,I_S\otimes\Lambda_j\otimes I_E]=0,\qquad j=\uparrow,\downarrow.}
$$
The environment interaction preserves these detector populations, making their basis a candidate <pointer basis>. It can change their relative coherence without flipping the stored labels.
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