  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 $\mathbb P^3$ constructed
in arXiv:1410.1585. Finally, we formulate a conjecture concerning the
contractibility of a principal connected component of $\mathrm{Stab}(\mathbb
P^3)$.
