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We construct a subset of the space of stability conditions for any projective threefold with an ample polarization that satisfies a certain Bogomolov-Gieseker inequality to refine the result in arXiv: 1410. 1585. Then, we demonstrate that the global dimension, as defined in arXiv: 2008. 00282 and arXiv: 1807. 00469, is 3 for any stability condition on P³ constructed in arXiv: 1410. 1585. Finally, we formulate a conjecture concerning the contractibility of a principal connected component of Stab (P³).
Wu et al. (Thu,) studied this question.