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The deep inside-outside recursive autoencoder (DIORA; Drozdov et al. 2019) is a self-supervised neural model that learns to induce syntactic tree structures for input sentences *without access to labeled training data*. In this paper, we discover that while DIORA exhaustively encodes all possible binary trees of a sentence with a soft dynamic program, its vector averaging approach is locally greedy and cannot recover from errors when computing the highest scoring parse tree in bottom-up chart parsing. To fix this issue, we introduce S-DIORA, an improved variant of DIORA that encodes a single tree rather than a softly-weighted mixture of trees by employing a hard argmax operation and a beam at each cell in the chart. Our experiments show that through *fine-tuning* a pre-trained DIORA with our new algorithm, we improve the state of the art in *unsupervised* constituency parsing on the English WSJ Penn Treebank by 2.2-6% F1, depending on the data used for fine-tuning.