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We revisit punctuation-aware tree binarization for constituency parsing and ask whether dependency-induced headedness improves binary parser supervision. Although learned heads substantially outperform rule-based heads in intrinsic head prediction, they do not yield consistent parsing gains after debinarization. In particular, punctuation-conditioned evaluation shows that learned headedness underperforms rule-based binarization in macro-average punctuation-sensitive $F_1$, despite a small overall gain on CTB. Similar instability appears under cross-treebank transfer. These results suggest that \ycc{linguistically grounded} headedness is not necessarily parser-optimal when used as a binarization control signal. The paper presents a negative result: better head prediction does not imply better punctuation-sensitive constituency parsing.