  The perfect cone compactification is a toroidal compactification which can be
defined for locally symmetric varieties. Let
$\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}$ be the perfect cone
compactification of the quotient of the type IV domain $D_{L}$ associated to an
even lattice $L$. In our main theorem we show that the pair ${
(\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}, \Delta/2) }$ has klt singularities,
where $\Delta$ is the closure of the branch divisor of ${
D_{L}/\widetilde{O}^{+}(L) }$. In particular this applies to the perfect cone
compactification of the moduli space of $2d$-polarised $K3$ surfaces with ADE
singularities when $d$ is square-free.
