KLT-singularities of horospherical pairs (after B.Pasquier, arXiv:1509.06502)

Let $X$ be a variety with an action of a reductive algebraic group $G$. Recall that $X$ is said to be horospherical if it is a fibration over a partial flag variety whose fiber is a smooth toric variety.
It turns out that for an effective $B$-invariant $\mathbb Q$-Cartier divisor $D$ on $X$, such that $D+K_X$ is also $\mathbb Q$-Cartier, the pair $(X,D)$ is Kawamata log-terminal iff $D=\sum a_i D_i$, with $D_i$ irreducible and $a_i\in [0,1)$.
The strategy of the proof is as follows: the case of horospherical $X$ can be reduced to the case of a flag variety. And if $X$ is a partial flag variety $G/P$, the klt condition can be reinterpreted combinatorially in terms of the root systems for $G$ and $P$, using Bott–Samelson desingularizations.