Decomposition of Frobenius pushforwards of line bundles on wonderful compactifications

De Concini and Procesi introduced varieties known as wonderful compactifications, which are smooth projective compactifications of semisimple adjoint groups G. We study the Frobenius pushforwards of line bundles on the wonderful compactifications, and in particular we decompose them into a direct sum of vector subbundles and explicitly describe the ranks. We are especially interested in when these subbundles are line bundles, and in the case of G=PSLn, we offer lower bounds on the multiplicities (as direct summands) for these line bundles.