  We compute projective dimension of translated simple modules in the regular
block of the BGG category $\mathcal{O}$ in terms of Kazhdan-Lusztig
combinatorics. This allows us to determine which projectives can appear at the
last step of a minimal projective resolution for a translated simple module,
confirming a conjecture by Johan K{\aa}hrstr\"om. We also derive some
inequalities, in terms of Lusztig's $\mathbf{a}$-function, for possible degrees
in which the top (or socle) of a translated simple module can live. Finally, we
relate Kostant's problem with decomposability and isomorphism of translated
simple modules, addressing yet another conjecture by Johan K{\aa}hrstr\"om.

Representation Theory