= Solution
Only the two lower orientations remain populated, giving $2^N$ equiprobable <microstates>. Thus
$$
\boxed{\lim_{T\to0}S=Nk_{\rm B}\ln2.}
$$
This <residual entropy> follows from the stipulated independent-site <ground-state degeneracy>. The zero-entropy version of the <third law of thermodynamics> presupposes a unique ordered <ground state>; that assumption does not hold here. Intermolecular interactions or a small splitting of the two in-plane states could remove this degeneracy, but they are excluded from the given model.
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