Generalization of the Frobenius-pushforward splitting theorem to semisimple adjoint groups

Let $G$ be a semisimple adjoint group, and let $X$ be the corresponding wonderful compactification. The divisorial and Frobenius-pushforward setup of Theorem result in the stated decomposition and splitting properties for $X$.

Generalization conjecture. Do the results of Theorem hold for any semisimple adjoint group $G$?

The conjecture proposes that the results established for the specific group considered in the theorem extend to all semisimple adjoint groups. The supplied text does not state whether this generalization is known or remains open.