  We investigate the existence of $\frac18$-BPS black hole microstates in the
$\mathfrak{su}(1,1|2)$ sector of Type IIB string theory on $\mathrm{AdS}_5
\times \mathrm{S}^5$. As will be explained, these states are in one-to-one
correspondence with the Schur operators comprising the chiral algebra of
$\mathcal{N}=4$ super-Yang-Mills, and a conjecture of Beem et al. implies that
the Schur sector only contains graviton operators and hence $\frac18$-BPS black
holes do not exist. We scrutinize this conjecture from multiple angles.
Concerning the macroscopic counting, we rigorously prove that the flavored
Schur index cannot exhibit black hole entropy growth, and provide numerical
evidence that the flavored MacDonald index also does not exhibit such growth.
Next, we go beyond counting to examine the algebraic structure, beginning by
presenting evidence for the well-definedness of the super-$\mathcal{W}$ algebra
of Beem et al., then using modular differential equations to argue for an upper
bound on the lightest non-graviton operator if existent, and finally performing
a systematic construction of cohomologies to recover only gravitons. Along the
way, we clarify key aspects of the 4d/2d correspondence using the formalism of
the holomorphic topological twist.
